Algebra I. Key Terms. Slide 1 / 175 Slide 2 / 175. Slide 3 / 175. Slide 4 / 175. Slide 5 / 175. Slide 6 / 175. Quadratics.

Size: px
Start display at page:

Download "Algebra I. Key Terms. Slide 1 / 175 Slide 2 / 175. Slide 3 / 175. Slide 4 / 175. Slide 5 / 175. Slide 6 / 175. Quadratics."

Transcription

1 Slide 1 / 175 Slide / 175 Algebra I Quadratics Key Terms Slide 3 / 175 Table of Contents Click on the topic to go to that section Slide 4 / 175 Characteristics of Quadratic Equations Transforming Quadratic Equations Graphing Quadratic Equations Solve Quadratic Equations by Graphing Solve Quadratic Equations by Factoring Key Terms Solve Quadratic Equations Using Square Roots Solve Quadratic Equations by Completing the Square The Discriminant Solve Quadratic Equations by Using the Quadratic Formula Solving Application Problems Slide 5 / 175 Axis of Symmetry Slide 6 / 175 Parabolas Return to Table of Contents Axis of symmetry: The vertical line that divides a parabola into two symmetrical halves Maximum: The y-value of the vertex if a < 0 and the parabola opens downward Minimum: The y-value of the vertex if a > 0 and the parabola opens upward Parabola: The curve result of graphing a quadratic equation (+ a) Min Max (- a)

2 Slide 7 / 175 Quadratics Slide 8 / 175 Quadratic Equation: An equation that can be written in the standard form ax + bx + c = 0. Where a, b and c are real numbers and a does not = 0. Vertex: The highest or lowest point on a parabola. Zero of a Function: An x value that makes the function equal zero. Characteristics of Quadratic Equations Return to Table of Contents Slide 9 / 175 Quadratics Slide 10 / 175 Writing Quadratic Equations A quadratic equation is an equation of the form ax + bx + c = 0, where a is not equal to 0. The form ax + bx + c = 0 is called the standard form of the quadratic equation. Practice writing quadratic equations in standard form: (Simplify if possible.) Write x = x + 4 in standard form: The standard form is not unique. For example, x - x + 1 = 0 can be written as the equivalent equation -x + x - 1 = 0. Also, 4x - x + = 0 can be written as the equivalent equation x - x + 1 = 0. Why is this equivalent? Slide 11 / 175 Slide 1 / Write 3x = -x + 7 in standard form: A. x + 3x-7= 0 B. x -3x +7=0 C. -x -3x -7= 0 Write 6x - 6x = 1 in standard form: A. 6x - 6x -1 = 0 B. x - x - = 0 C. -x + x + = 0

3 Slide 13 / Write 3x - = 5x in standard form: A. x + = 0 B. -x - = 0 Slide 14 / 175 Characteristics of Quadratic Functions The graph of a quadratic is a parabola, a u-shaped figure. The parabola will open upward or downward. C. not a quadratic equation upward downward Slide 15 / 175 Characteristics of Quadratic Functions A parabola that opens upward contains a vertex that is a minimum point. A parabola that opens downward contains a vertex that is a maximum point. Slide 16 / 175 Characteristics of Quadratic Functions The domain of a quadratic function is all real numbers. vertex D = Reals vertex Slide 17 / 175 Characteristics of Quadratic Functions To determine the range of a quadratic function, ask yourself two questions: > Is the vertex a minimum or maximum? > What is the y-value of the vertex? Slide 18 / 175 Characteristics of Quadratic Functions If the vertex is a maximum, then the range is all real numbers less than or equal to the y-value. The range of this quadratic is (,10] If the vertex is a minimum, then the range is all real numbers greater than or equal to the y-value. The range of this quadratic is [ 6, )

4 Slide 19 / 175 Characteristics of Quadratic Functions Slide 0 / 175 Characteristics of Quadratic Functions 7. An axis of symmetry (also known as a line of symmetry) will divide the parabola into mirror images. The line of symmetry is always a vertical line of the form To find the axis of symmetry simply plug the values of a and b into the equation: Remember the form ax + bx + c. In this example a =, b = -8 and c = x = b a y = x 8x + x = ( 8) () = x= a b c y = x 8x + x = b a x = ( 8) () = x= Slide 1 / 175 Characteristics of Quadratic Functions The x-intercepts are the points at which a parabola intersects the x-axis. These points are also known as zeroes, roots or solutions and solution sets. Each quadratic equation will have two, one or no real x-intercepts. Slide / The vertical line that divides a parabola into two symmetrical halves is called... A discriminant B perfect square C axis of symmetry D vertex E slice Slide 3 / 175 Slide 4 / The equation y = x + 3x 18 is graphed on the set of axes below. Based on this graph, what are the roots of the equation x + 3x 18 = 0? A B C D 3 and 6 0 and 18 3 and 6 3 and 18 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 011.

5 Slide 5 / The equation y = x x + 8 is graphed on the set of axes below. Based on this graph, what are the roots of the equation x x + 8 = 0? A 8 and 0 B and 4 C 9 and 1 D 4 and Slide 6 / What is an equation of the axis of symmetry of the parabola represented by y = x + 6x 4? A x = 3 B y = 3 C x = 6 D y = 6 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 011. From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 011. Slide 7 / 175 Slide 8 / The height, y, of a ball tossed into the air can be represented by the equation y = x + 10x + 3, where x is the elapsed time. What is the equation of the axis of symmetry of this parabola? A y = 5 B y = 5 C x = 5 D x = 5 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 011. Slide 9 / 175 Transforming Quadratic Equations Return to Table of Contents Slide 30 / 175 Quadratic Parent Equation y = x 3 The quadratic parent equation is y = x. The graph of all other quadratic equations are transformations of the graph of y= x. x x y = x

6 Slide 31 / 175 Quadratic Parent Equation The quadratic parent equation is y = x. How is y = x changed into y = x? Slide 3 / 175 Quadratic Parent Equation The quadratic parent equation is y = x. How is y = x changed into y =.5x? y = x x y = x x y = x y = x Slide 33 / 175 What Does "A" Do? What does "a" do in y = ax + bx + c? How does a > 0 affect the parabola? How does a < 0 affect the parabola? y = x y = x y = Slide 34 / 175 What Does "A" Do? What does "a" also do in y =ax + bx +c? How does your conclusion about "a" change as "a" changes? 1 x y = 3x y = x Slide 35 / y = 1x y = 3x y = x Slide 36 / 175 What Does "A" Do? What does "a" do in y = ax + bx + c? 11 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent equation. If a > 0, the graph opens up. If a < 0, the graph opens down. If the absolute value of a is > 1, then the graph of the equation is narrower than the graph of the parent equation. If the absolute value of a is < 1, then the graph of the equation is wider than the graph of the parent equation. A B C D y =.3x up, wider up, narrower down, wider down, narrower

7 Slide 37 / Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent function. y = 4x Slide 38 / Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent equation. y = x + 100x + 45 A up, wider A up, wider B up, narrower B up, narrower C down, wider C down, wider D down, narrower D down, narrower Slide 39 / 175 Slide 40 / Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent function. 15 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent function. A up, wider y = x 3 A up, wider y = 7 x 5 B up, narrower B up, narrower C down, wider C down, wider D down, narrower D down, narrower Slide 41 / 175 Slide 4 / 175 What Does "C" Do? What Does "C" Do? What does "c" do in y = ax + bx + c? What does "c" do in y = ax + bx + c? y = x + 6 y = x + 3 y = x "c" moves the graph up or down the same value as "c." "c" is the y- intercept. y = x y = x 5 y = x 9

8 Slide 43 / Without graphing, what is the y- intercept of the the given parabola? y = x + 17 Slide 44 / Without graphing, what is the y- intercept of the the given parabola? y = x 6 Slide 45 / Without graphing, what is the y- intercept of the the given parabola? y = 3x + 13x 9 Slide 46 / Without graphing, what is the y- intercept of the the given parabola? y = x + 5x Slide 47 / Choose all that apply to the following quadratic: Slide 48 / Choose all that apply to the following quadratic: A B C D f(x) =.7x 4 opens up opens down wider than parent function narrower than parent function A y-intercept of y = 4 B y-intercept of y = C y-intercept of y = 0 D y-intercept of y = E y-intercept of y = 4 A B C D opens up opens down wider than parent function narrower than parent function f(x) = 4 x 6x 3 E y-intercept of y = 4 F y-intercept of y = G y-intercept of y = 0 H y-intercept of y = I y-intercept of y = 4 J y-intercept of y = 6 F y-intercept of y = 6

9 Slide 49 / 175 Slide 50 / 175 Graphing Quadratic Equations Return to Table of Contents Slide 51 / 175 Graph by Following Six Steps: Step 1 - Find Axis of Symmetry Slide 5 / 175 Axis of Symmetry Step 1 - Find Axis of Symmetry What is the Axis of Symmetry? Step - Find Vertex Step 3 - Find Y intercept Step 4 - Find two more points Axis of Symmetry Step 5 - Partially graph Step 6 - Reflect Slide 53 / 175 Slide 54 / 175

10 Slide 55 / 175 Step 3 - Find y intercept What is the y-intercept? Slide 56 / 175 Step 3 - Find y intercept Graph y = 3x 6x + 1 The y- intercept is always the c value, because x = 0. y- intercept y = ax + bx + c y = 3x 6x + 1 c = 1 The y-intercept is 1 and the graph passes through (0,1). Slide 57 / 175 Step 4 - Find Two More Points Graph y = 3x 6x + 1 Find two more points on the parabola. Slide 58 / 175 Step 4 - Find Two More Points (continued) Graph y = 3x 6x + 1 Choose different values of x and plug in to find points. Let's pick x = 1 and x = y = 3x 6x + 1 y = 3( 1) 6( 1) + 1 y = y = 10 ( 1,10) y = 3x 6x + 1 y = 3( ) 6( ) + 1 y = 3(4) y = 5 (, 5) Slide 59 / 175 Step 5 - Graph the Axis of Symmetry Slide 60 / 175 Step 6 - Reflect the Points Graph the axis of symmetry, the vertex, the point containing the y-intercept and two other points. Reflect the points across the axis of symmetry. Connect the points with a smooth curve. (4,5)

11 Slide 61 / What is the axis of symmetry for y = x + x - 3 (Step 1)? Slide 6 / What is the vertex for y = x + x - 3 (Step )? A (-1, -4) B (1, -4) C (-1, 4) Slide 63 / What is the y-intercept for y = x + x - 3 (Step 3)? A -3 B 3 axis of symmetry = 1 vertex = 1, 4 y intercept = 3 other points (step 4) (1,0) (,5) Partially graph (step 5) Reflect (step 6) Slide 64 / 175 Graph y= x + x 3 Slide 65 / 175 Graph y = x 6x + 4 Slide 66 / 175 Graph y = x 4x + 5

12 Slide 67 / 175 Slide 68 / 175 Graph y = 3x 7 Solve Quadratic Equations by Graphing Return to Table of Contents Slide 69 / 175 Find the Zeros One way to solve a quadratic equation in standard form is find the zeros by graphing. Slide 70 / 175 Find the Zeros How many zeros do the parabolas have? What are the values of the zeros? A zero is the point at which the parabola intersects the x-axis. A quadratic may have one, two or no zeros. clickno zeroes (doesn't cross the "x" axis) click zeroes; x = -1 and x=3 click 1 zero; x=1 Slide 71 / 175 Review To solve a quadratic equation by graphing follow the 6 step process we already learned. 6 Slide 7 / 175 Solve the equation by graphing. 1x + 18 = x Which of these is in standard form? Step 1 - Find Axis of Symmetry Step - Find Vertex Step 3 - Find Y intercept Step 4 - Find two more points Step 5 - Partially graph Step 6 - Reflect A B C y = x 1x + 18 y = x 1x + 18 y = x + 1x 18

13 Slide 73 / 175 Slide 74 / What is the axis of symmetry? y = x + 1x 18 A 3 B 3 C 4 D 5 8 y = x + 1x 18 What is the vertex? A (3,0) B ( 3,0) C (4,0) D ( 5,0) Slide 75 / 175 Slide 76 / y = x + 1x 18 What is the y- intercept? 30 A If two other points are (5, 8) and (4, ),what does the graph of y = x + 1x 18 look like? B A (0, 0) B (0, 18) C (0, 18) D (0, 1) C D Slide 77 / 175 Slide 78 / y = x + 1x 18 What is(are) the zero(s)? A 18 B 4 C 3 D 8 click for graph of answer Solve Quadratic Equations by Factoring Return to Table of Contents

14 Slide 79 / 175 Solving Quadratic Equations by Factoring Slide 80 / 175 Solving Quadratic Equations by Factoring Review of factoring - To factor a quadratic trinomial of the form x + bx + c, find two factors of c whose sum is b. Example - To factor x + 9x + 18, look for factors whose sum is 9. Factors of 18 Sum 1 and and and 6 9 x + 9x + 18 = (x + 3)(x + 6) When c is positive, it's factors have the same sign. The sign of b tells you whether the factors are positive or negative. When b is positive, the factors are positive. When b is negative, the factors are negative. 1. Multiply the First terms (x + 3)(x + ) x x = x. Multiply the Outer terms (x + 3)(x + ) x = x 3. Multiply the Inner terms (x + 3)(x + ) 3 x = 3x 4. Multiply the Last terms (x + 3)(x + ) 3 = 6 Slide 81 / 175 Solving Quadratic Equations by Factoring Remember the FOIL method for multiplying binomials Slide 8 / 175 Zero Product Property For all real numbers a and b, if the product of two quantities equals zero, at least one of the quantities equals zero. Numbers Algebra 3(0) = 0 If ab = 0, 4(0) = 0 Then a = 0 or b = 0 (x + 3)(x + ) = x + x + 3x + 6 = x + 5x + 6 F O I L Slide 83 / 175 Zero Product Property Example 1: Solve x + 4x 1 = 0 (x + 6) (x ) = 0 Use "FUSE"! Factor the trinomial using the FOIL method. Example : Solve x + 36 = 1x 1x 1x Slide 84 / 175 Zero Product Property x 1x + 36 = 0 The equation has to be written in standard form (ax + bx + c). So subtract 1x from both sides. x + 6 = 0 or x = x = 6 x = Use the Zero property (x 6)(x 6) = 0 x 6 = Factor the trinomial using the FOIL method ( 6) 1 = ( 4) 1 = = 0 0 = 0 or + 4() 1 = = 0 0 = 0 Substitue found value into original equation Equal - problem solved! The solutions are -6 and. x = = 1(6) = 7 7 = 7 Use the Zero property Substitue found value into original equation Equal - problem solved!

15 Slide 85 / 175 Example 3: Solve x 16x + 48= 0 (x 4)(x 1) = 0 Zero Product Property x 4 = 0 x 1 = x = 4 x = (4) + 48 = = = 0 0 = 0 Factor the trinomial using the FOIL method. Use the Zero property Substitue found value into original equation 3 Solve x 5x + 6 = 0 A 7 F 3 B 5 G 5 C 3 H 6 D I 7 E J 15 Slide 86 / (1) + 48 = = = 0 0 = 0 48 Equal - problem solved! Slide 87 / 175 Slide 88 / Solve m + 10m + 5 = 0 34 Solve h h = 1 A 7 B 5 C 3 D E F 3 G 5 H 6 I 7 J 15 A 1 B 4 C 3 D E F 3 G 4 H 6 I 8 J 1 Slide 89 / 175 Slide 90 / Solve d 35d = d 36 Solve 8y + y = 3 A 7 F 0 A 3 / 4 F 3 / 4 B 5 G 5 B 1 / G 1 / C 3 H 6 C 4 / 3 H 4 / 3 D 35 E 1 I 7 J 37 D E I 3 J 3

16 Slide 91 / 175 Slide 9 / Which equation has roots of 3 and 5? A x + x 15 = 0 B x x 15 = 0 C x + x + 15 = 0 D x x + 15 = 0 Solve Quadratic Equations Using Square Roots Return to Table of Contents Slide 93 / 175 Square Root Method Slide 94 / 175 Square Root Method You can solve a quadratic equation by the square root method if you can write it in the form: x² = c If x and c are algebraic expressions, then: Solve for z: z² = 49 z = ± 49 z = ±7 x = c written as: x = ± c or x = c The solution set is 7 and 7 Slide 95 / 175 Slide 96 / 175 A quadratic equation of the form x = c can be solved using the Square Root Property. Example: Solve 4x = 0 Square Root Method 4x = x = 5 Divide both sides by 4 to isolate x² The solution set is 5 and 5 Square Root Method Solve 5x² = 0 using the square root method: 5x = x = 4 x = 4 or x = 4 x = ± x = ± 5

17 Slide 97 / 175 Square Root Method Solve (x 1)² = 0 using the square root method. Slide 98 / When you take the square root of a real number, your answer will always be positive. click x 1 = 0 x 1 = (4)(5) x 1 = 5 x = x = or click x 1 = 0 x 1 = (4)(5) x 1 = 5 x = x = True False solution: x = 1 ± 5 click Slide 99 / 175 Slide 100 / If x = 16, then x = A 4 B C D 6 E 4 40 If y = 4, then y = A 4 B C D 6 E 4 Slide 101 / 175 Slide 10 / If 8j = 96, then j = A 3 B 3 C 3 D 3 E ±1 4 If 4h 10= 30, then h = A 10 B 5 C 5 D 10 E ±10

18 Slide 103 / 175 Slide 104 / If (3g 9) + 7= 43, then g = A 1 B C D 5 Solving Quadratic Equations by Completing the Square E ±3 Return to Table of Contents Slide 105 / 175 Find the Missing Value of "C" 44 Find ( b / ) if b = 14 Slide 106 / 175 Before we can solve the quadratic equation, we first have to find the missing value of C. To do this, simply take the value of b, divide it in and then square the result. ax +bx+c (b/) Find the value that completes the square. 8/ = 4 4 = 16 x + 8x + x + 0x x 16x + 64 x x Find ( b / ) if b = 1 Slide 107 / 175 Slide 108 / Complete the square to form a perfect square trinomial x + 18x +?

19 Slide 109 / Complete the square to form a perfect square trinomial x 6x +? Slide 110 / 175 Solving Quadratic Equations by Completing the Square Step 1 - Write the equation in the form x + bx = c Step - Find (b ) Step 3 - Complete the square by adding (b ) to both sides of the equation. Step 4 - Factor the perfect square trinomial. Step 5 - Take the square root of both sides Step 6 - Write two equations, using both the positive and negative square root and solve each equation. Slide 111 / 175 Let's look at an example to solve: x + 14x = 15 x + 14x = 15 Solving Quadratic Equations by Completing the Square Step 1 - Already done! (14 ) = 49 Step - Find (b ) x + 14x + 49 = Step 3 - Add 49 to both sides (x + 7) = 64 Step 4 - Factor and simplify x x = x x = Slide 11 / 175 Solving Quadratic Equations by Completing the Square Another example to solve: x x = 0 Step 1 - Write as x +bx=c ( ) = ( 1) = 1 Step - Find (b ) x x + 1 = + 1 Step 3 - Add 1 to both sides (x 1) = 3 Step 4 - Factor and simplify x + 7 = ±8 Step 5 - Take the square root of both sides x 1 = ± 3 Step 5 - Take the square root of both sides x + 7 = 8 or x + 7 = 8 x = 1 or x = 15 Step 6 - Write and solve two equations x 1 = 3 or x 1 = 3 x = or x = 1 3 Step 6 - Write and solve two equations Slide 113 / Solve the following by completing the square : x + 6x = 5 A 5 B C 1 D 5 E Slide 114 / Solve the following by completing the square: x 8x = 0 A 10 B C 1 D 10 E

20 Slide 115 / Solve the following by completing the square : 36x = 3x A 6 B 6 C 0 D 6 E 6 A more difficult example: 5 4 x = ± 3 3 Slide 116 / 175 Solve 3x 10x = 3 3x 10x = x x = = 10 x 1 = 5 = 5 ) ( ( ) ( ) 10x 5 5 x + = ( ) 5 16 x = 3 9 Write as x +bx=c Find (b ) Add 5/9 to both sides Factor and simplify Take the square root of both sides Slide 117 / 175 Slide 118 / Solve the following by completing the square: 4x 7x = 0 A 1 4 B C The Discriminant D E 5 4 Return to Table of Contents Slide 119 / 175 The Discriminant Slide 10 / 175 The Discriminant Discriminant - the part of the equation under the radical sign in a quadratic equation. x = b ± b 4ac a ax + bx + c = 0 The discriminant, b 4ac, or the part of the equation under the radical sign, may be used to determine the number of real solutions there are to a quadratic equation. b 4ac is the discriminant If b 4ac > 0, the equation has two real solutions If b 4ac = 0, the equation has one real solution If b 4ac < 0, the equation has no real solutions

21 Slide 11 / 175 The Discriminant Slide 1 / 175 The Discriminant Remember: The square root of a positive number has two solutions. The square root of zero is 0. The square root of a negative number has no real solution. Example 4 = ± () () = 4 and ( )( ) = 4 So BOTH and are solutions Slide 13 / 175 The Discriminant What is the relationship between the discriminant of a quadratic and its graph? Slide 14 / 175 The Discriminant What is the relationship between the discriminant of a quadratic and its graph? y = x 4x + y = x + 6x + 9 y = x 8x + 10 y = 3x + 8x 4 Discriminant (8) 4(1)(10) = = 4 ( 6) 4(3)( 4) = = 84 Slide 15 / 175 The Discriminant What is the relationship between the discriminant of a quadratic and its graph? Discriminant ( 4) 4()() = = 0 (6) 4(1)(9) = = 0 Slide 16 / What is value of the discriminant of x 3x + 5 = 0? y = x + 5x + 9 y = 3x 3x + 4 Discriminant (5) 4(1)(9) = 5 36 = 11 ( 3) 4(3)(4) = 9 48 = 39

22 Slide 17 / Find the number of solutions using the discriminant for x 3x + 5 = 0 A 0 Slide 18 / What is value of the discriminant of x 8x + 4 = 0? B 1 C Slide 19 / 175 Slide 130 / Find the number of solutions using the discriminant for x 8x + 4 = 0 A 0 B 1 C Solve Quadratic Equations by Using the Quadratic Formula Return to Table of Contents Slide 131 / 175 Solve Any Quadratic Equation Slide 13 / 175 The Quadratic Formula At this point you have learned how to solve quadratic equations by: graphing factoring using square roots and completing the square Many quadratic equations may be solved using these methods; however, some cannot be solved using any of these methods. The solutions of ax + bx + c = 0, where a 0, are: x = b ± b 4ac a "x equals the opposite of b, plus or minus the square root of b squared minus 4ac, all divided by a." Today we will be given a tool to solve ANY quadratic equation. It ALWAYS works.

23 Slide 133 / 175 The Quadratic Formula Slide 134 / 175 The Quadratic Formula Example 1 x + 3x 5 = 0 x + 3x + ( 5) = 0 Identify values of a, b and c x = 3 ± 9 ( 40) 4 Simplify x = b ± b 4ac a Write the Quadratic Formula x = 3 ± 49 4 = 3 ± 7 4 Write as two equations x = 3 ± 3 4()( 5) () Substitute the values of a, b and c x = x = or 4 continued on next slide x = 1 or x = 5 Solve each equation Slide 135 / 175 The Quadratic Formula Example x = x 3 x + ( x) + ( 3) = 0 Slide 136 / 175 The Quadratic Formula Remember - In order to use the Quadratic Formula, the equation must be in standard form (ax + bx +c = 0). First, rewrite the equation in standard form. x = x 3 x x 0 = x + (-x) + ( 3) Use only addition for standard form 1x + ( x) + ( 3) = 0 x = b ± b 4ac a x = ( ) ± ( ) 4(1)( 3) (1) Identify values of a, b and c Write the Quadratic Formula Substitute the values of a, b and c x + ( x) + ( 3) = 0 Flip the equation Continued on next slide Now you are ready to use the Quadratic Formula Solution on next slide Slide 137 / 175 The Quadratic Formula Slide 138 / Solve the following equation using the quadratic formula: x = ± 4 ( 1) a x = ± 16 = ± 4 x = ± 4 or x = - 4 x = 3 or x = 1 Simplify Write as two equations Solve each equation x 5x + 4 = 0 A -5 F 1 B -4 G C -3 H 3 D - I 4 E -1 J 5

24 Slide 139 / Solve the following equation using the quadratic formula: x = x + 0 Slide 140 / Solve the following equation using the quadratic formula: x + 1 = 11x A 5 B 4 C 3 D E 1 F 1 G H 3 I 4 J 5 A 5 B 4 3 C D E 1 F 1 G 3 H I 4 J 5 Example 3 x x 4 = 0 1x + ( x) + ( 4) = 0 x = -b ± b -4ac a x = ( ) ± ( ) 4(1)( 4) Slide 141 / 175 The Quadratic Formula (1) Identify values of a, b and c Write the Quadratic Formula Substitute the values of a, b and c Slide 14 / 175 The Quadratic Formula x = ± 4 ( 16) Simplify x = ± 0 x = ± 0 or x = - 0 x = ± 5 or x = - 5 Write as two equations x = or x = 1 5 Continued on next slide x 3.4 or x 1.4 Use a calculator to estimate x 59 Find the larger solution to x + 6x 1 = 0 Slide 143 / Find the smaller solution to x + 6x 1 = 0 Slide 144 / 175

25 Slide 145 / 175 Slide 146 / 175 Quadratic Equations and Applications A sampling of applied problems that lend themselves to being solved by quadratic equations: Application Problems Number Reasoning Distances Geometry: Dimensions Free Falling Objects Height of a Projectile Return to Table of Contents Slide 147 / 175 The product of two consecutive negative integers is 1,1. What are the numbers? Remember that consecutive integers are one unit apart, so the numbers are n and n + 1. Multiplying to get the product: n(n + 1) = 11 n + n = 11 n + n 11 = 0 (n + 34)(n - 33) = 0 Number Reasoning STANDARD Form FACTOR Slide 148 / 175 Application Problems PLEASE KEEP THIS IN MIND When solving applied problems that lead to quadratic equations, you might get a solution that does not satisfy the physical constraints of the problem. For example, if x represents a width and the two solutions of the quadratic equations are 9 and 1, the value 9 is rejected since a width must be a positive number. n = 34 and n = 33. The solution is either 34 and 33 or 33 and 34, since the direction ask for negative integers 34 and 33 are the correct pair. Slide 149 / The product of two consecutive even integers is 48. Find the smaller of the two integers. Hint: x(x+) = 48 Click to reveal hint Slide 150 / 175 Application Problems TRY THIS: The product of two consecutive integers is 7. What are the numbers?

26 6 Slide 151 / 175 The product of two consecutive even integers is 58. What is the smaller number? Slide 15 / 175 More of a challenge... The product of two consecutive odd integers is 1 less than four times their sum. Find the two integers. Let n = 1st number n + = nd number n(n + ) = 4[n + (n + )] 1 n + n = 4[n + ] 1 n + n = 8n n + n = 8n + 7 n 6n - 7 = 0 (n 7)(n + 1) = 0 n = 7 and n = 1 Which one do you use? Or do you use both? Slide 153 / 175 More of a challenge... If n = 7 then n + = 9 Slide 154 / The product of a number and a number 3 more than the original is 418. What is the smallest value the original number can be? 7 x 9 = 4[7 + (7 + )] 1 63 = 4(16) 1 63 = = 63 If n = 1 then n + = 1 + = 1 ( 1) x 1 = 4[ 1 + ( 1 + )] 1 1 = 4[ 1 + 1] 1 1 = 4(0) 1 1 = 1 We get two sets of answers. Slide 155 / 175 Slide 156 / Find three consecutive positive even integers such that the product of the second and third integers is twenty more than ten times the first integer. Enter the value of the smaller even integer. 65 When 36 is subtracted from the square of a number, the result is five times the number. What is the positive solution? A 9 B 6 C 3 D 4 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 011. From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 011.

27 Slide 157 / Tamara has two sisters. One of the sisters is 7 years older than Tamara.The other sister is 3 years younger than Tamara. The product of Tamara s sisters ages is 4. How old is Tamara? From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 011. Step 5 - Solve a + b = c x + (x+0) = 100 x + x + 40x = 10,000 x + 40x 9600 = 0 (x +0x 4800) = 0 x + 0x 4800 = 0 Slide 159 / 175 Distance Problems 100 x x+0 Square the binomial Standard form Factor Divide each side by Think about your options for solving the rest of this equation. Completing the square? Quadratic Formula? Continued on next slide Slide 158 / 175 Distance Problems Example Two cars left an intersection at the same time, one heading north and one heading west. Some time later, they were exactly 100 miles apart. The car that headed north had gone 0 miles farther than the car headed west. How far had each car traveled? Step 1 - Read the problem carefully. Step - Illustrate or draw your information. 100 x+0 Step 3 - Assign a variable Let x = the distance traveled by the car heading x west Then (x + 0) = the distance traveled by the car heading north Step 4 - Write an equation Does your drawing remind you of the Pythagorean Theorem? a + b = c Continued on next slide x = 0 ± 400 4(1)( 4800) Slide 160 / 175 Distance Problems Did you try the quadratic formula? x = 0 ± 19,600 x = 60 or x = -80 Since the distance cannot be negative, discard the negative solution. The distances are 60 miles and = 80 miles. Step 6 - Check your answers. Slide 161 / Two cars left an intersection at the same time,one heading north and the other heading east. Some time later they were 00 miles apart. If the car heading east traveled 40 miles farther, how far did the car traveling north go? Slide 16 / 175 Geometry Applications Area Problem The length of a rectangle is 6 inches more than its width. The area of the rectangle is 91 square inches. Find the dimensions of the rectangle. Step 1 - Draw the picture of the rectangle. Let the width = x and the length = x + 6 Step - Write the equation using the formula Area = length x width x + 6 x

28 Slide 163 / 175 Geometry Applications Step 3 - Solve the equation Slide 164 / The width of a rectangular swimming pool is 10 feet less than its length. The surface area of the pool is 600 square feet. What is the pool's width? x( x + 6) = 91 x + 6x = 91 x + 6x 91 = 0 Hint: (L)(L 10) = 600. Click to reveal hint (x 7)(x + 13) = 0 x = 7 or x = 13 Since a length cannot be negative... The width is 7 and the length is 13. Slide 165 / 175 Slide 166 / A square's length is increased by 4 units and its width is increased by 6 units. The result of this transformation is a rectangle with an area that 195 square units. Find the area of the original square. 70 The rectangular picture frame below is the same width all the way around. The photo it surrounds measures 17" by 11". The area of the frame and photo combined is 315 sq. in. What is the length of the outer frame? length x x Slide 167 / 175 Slide 168 / The area of the rectangular playground enclosure at South School is 500 square meters. The length of the playground is 5 meters longer than the width. Find the dimensions of the playground, in meters. [Only an algebraic solution will be accepted.] 7 Jack is building a rectangular dog pen that he wishes to enclose. The width of the pen is yards less than the length. If the area of the dog pen is 15 square yards, how many yards of fencing would he need to completely enclose the pen? From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 011.

29 Slide 169 / 175 Slide 170 / A person walking across a bridge accidentally drops an orange in the river below from a height of 40 ft. The function h = 16t + 40 gives the orange's approximate height h above the water, in feet, after t seconds. In how many t seconds will the orange hit the water? (Round to the nearest tenth.) Free Falling Objects Problems Hint: when it hits the water it is at 0. Slide 171 / Greg is in a car at the top of a roller-coaster ride. The distance, d, of the car from the ground as the car descends is determined by the equation d = t where t is the number of seconds it takes the car to travel down to each point on the ride. How many seconds will it take Greg to reach the ground? Slide 17 / The height of a golf ball hit into the air is modeled by the equation h = 16t + 48t, where h represents the height, in feet, and t represents the number of seconds that have passed since the ball was hit. What is the height of the ball after seconds? A 16 ft B 3 ft C 64 ft D 80 ft From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 011. Slide 173 / 175 Slide 174 / A skyrocket is shot into the air. It's altitude in feet, h, after t seconds is given by the function h = 16t + 18t. What is the rocket's maximum altitude? Height of Projectiles Problems

30 Slide 175 / A rocket is launched from the ground and follows a parabolic path represented by the equation y = x + 10x. At the same time, a flare is launched from a height of 10 feet and follows a straight path represented by the equation y = x Using the accompanying set of axes, graph the equations that represent the paths of the rocket and the flare, and find the coordinates of the point or points where the paths intersect. From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 011.

Algebra I. Slide 1 / 175. Slide 2 / 175. Slide 3 / 175. Quadratics. Table of Contents Key Terms

Algebra I. Slide 1 / 175. Slide 2 / 175. Slide 3 / 175. Quadratics. Table of Contents Key Terms Slide 1 / 175 Slide 2 / 175 Algebra I Quadratics 2015-11-04 www.njctl.org Key Terms Table of Contents Click on the topic to go to that section Slide 3 / 175 Characteristics of Quadratic Equations Transforming

More information

Algebra I Quadratics

Algebra I Quadratics 1 Algebra I Quadratics 2015-11-04 www.njctl.org 2 Key Terms Table of Contents Click on the topic to go to that section Characteristics of Quadratic Equations Transforming Quadratic Equations Graphing Quadratic

More information

Quadratic Functions. Key Terms. Slide 1 / 200. Slide 2 / 200. Slide 3 / 200. Table of Contents

Quadratic Functions. Key Terms. Slide 1 / 200. Slide 2 / 200. Slide 3 / 200. Table of Contents Slide 1 / 200 Quadratic Functions Table of Contents Key Terms Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic Equations

More information

Quadratic Functions. Key Terms. Slide 2 / 200. Slide 1 / 200. Slide 3 / 200. Slide 4 / 200. Slide 6 / 200. Slide 5 / 200.

Quadratic Functions. Key Terms. Slide 2 / 200. Slide 1 / 200. Slide 3 / 200. Slide 4 / 200. Slide 6 / 200. Slide 5 / 200. Slide 1 / 200 Quadratic Functions Slide 2 / 200 Table of Contents Key Terms Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic

More information

Slide 1 / 200. Quadratic Functions

Slide 1 / 200. Quadratic Functions Slide 1 / 200 Quadratic Functions Key Terms Slide 2 / 200 Table of Contents Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic

More information

Algebra I Quadratic & Non-Linear Functions

Algebra I Quadratic & Non-Linear Functions 1 Algebra I Quadratic & Non-Linear Functions 2015-11-04 www.njctl.org 2 Table of Contents Click on the topic to go to that section Key Terms Explain Characteristics of Quadratic Functions Graphing Quadratic

More information

3.4 Solving Quadratic Equations by Completing

3.4 Solving Quadratic Equations by Completing www.ck1.org Chapter 3. Quadratic Equations and Quadratic Functions 3.4 Solving Quadratic Equations by Completing the Square Learning objectives Complete the square of a quadratic expression. Solve quadratic

More information

3.4 Solving Quadratic Equations by Completing

3.4 Solving Quadratic Equations by Completing .4. Solving Quadratic Equations by Completing the Square www.ck1.org.4 Solving Quadratic Equations by Completing the Square Learning objectives Complete the square of a quadratic expression. Solve quadratic

More information

Quadratic Equations Chapter Questions

Quadratic Equations Chapter Questions Quadratic Equations Chapter Questions 1. Describe the characteristics of a quadratic equation. 2. What are the steps for graphing a quadratic function? 3. How can you determine the number of solutions

More information

Quadratic Functions and Equations

Quadratic Functions and Equations Quadratic Functions and Equations Quadratic Graphs and Their Properties Objective: To graph quadratic functions of the form y = ax 2 and y = ax 2 + c. Objectives I can identify a vertex. I can grapy y

More information

2 P a g e. Essential Questions:

2 P a g e. Essential Questions: NC Math 1 Unit 5 Quadratic Functions Main Concepts Study Guide & Vocabulary Classifying, Adding, & Subtracting Polynomials Multiplying Polynomials Factoring Polynomials Review of Multiplying and Factoring

More information

Algebra II Unit #2 4.6 NOTES: Solving Quadratic Equations (More Methods) Block:

Algebra II Unit #2 4.6 NOTES: Solving Quadratic Equations (More Methods) Block: Algebra II Unit # Name: 4.6 NOTES: Solving Quadratic Equations (More Methods) Block: (A) Background Skills - Simplifying Radicals To simplify a radical that is not a perfect square: 50 8 300 7 7 98 (B)

More information

Name Date Class California Standards 17.0, Quadratic Equations and Functions. Step 2: Graph the points. Plot the ordered pairs from your table.

Name Date Class California Standards 17.0, Quadratic Equations and Functions. Step 2: Graph the points. Plot the ordered pairs from your table. California Standards 17.0, 1.0 9-1 There are three steps to graphing a quadratic function. Graph y x 3. Quadratic Equations and Functions 6 y 6 y x y x 3 5 1 1 0 3 1 1 5 0 x 0 x Step 1: Make a table of

More information

6.1 Quadratic Expressions, Rectangles, and Squares. 1. What does the word quadratic refer to? 2. What is the general quadratic expression?

6.1 Quadratic Expressions, Rectangles, and Squares. 1. What does the word quadratic refer to? 2. What is the general quadratic expression? Advanced Algebra Chapter 6 - Note Taking Guidelines Complete each Now try problem in your notes and work the problem 6.1 Quadratic Expressions, Rectangles, and Squares 1. What does the word quadratic refer

More information

MAHS-DV Algebra 1-2 Q4

MAHS-DV Algebra 1-2 Q4 MAHS-DV Algebra 1-2 Q4 Adrienne Wooten Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org To access a customizable version of this book, as well as other interactive

More information

Unit 5 Test: 9.1 Quadratic Graphs and Their Properties

Unit 5 Test: 9.1 Quadratic Graphs and Their Properties Unit 5 Test: 9.1 Quadratic Graphs and Their Properties Quadratic Equation: (Also called PARABOLAS) 1. of the STANDARD form y = ax 2 + bx + c 2. a, b, c are all real numbers and a 0 3. Always have an x

More information

Chapter 5 Smartboard Notes

Chapter 5 Smartboard Notes Name Chapter 5 Smartboard Notes 10.1 Graph ax 2 + c Learning Outcome To graph simple quadratic functions Quadratic function A non linear function that can be written in the standard form y = ax 2 + bx

More information

Section 1.1. Chapter 1. Quadratics. Parabolas. Example. Example. ( ) = ax 2 + bx + c -2-1

Section 1.1. Chapter 1. Quadratics. Parabolas. Example. Example. ( ) = ax 2 + bx + c -2-1 Chapter 1 Quadratic Functions and Factoring Section 1.1 Graph Quadratic Functions in Standard Form Quadratics The polynomial form of a quadratic function is: f x The graph of a quadratic function is a

More information

4.1 Graphical Solutions of Quadratic Equations Date:

4.1 Graphical Solutions of Quadratic Equations Date: 4.1 Graphical Solutions of Quadratic Equations Date: Key Ideas: Quadratic functions are written as f(x) = x 2 x 6 OR y = x 2 x 6. f(x) is f of x and means that the y value is dependent upon the value of

More information

ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION (DAY 1)

ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION (DAY 1) ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION (DAY 1) The Quadratic Equation is written as: ; this equation has a degree of. Where a, b and c are integer coefficients (where a 0)

More information

Quadratic Equations and Quadratic Functions

Quadratic Equations and Quadratic Functions Quadratic Equations and Quadratic Functions Eve Rawley, (EveR) Anne Gloag, (AnneG) Andrew Gloag, (AndrewG) Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access

More information

Chapter 2 Polynomial and Rational Functions

Chapter 2 Polynomial and Rational Functions SECTION.1 Linear and Quadratic Functions Chapter Polynomial and Rational Functions Section.1: Linear and Quadratic Functions Linear Functions Quadratic Functions Linear Functions Definition of a Linear

More information

= 9 = x + 8 = = -5x 19. For today: 2.5 (Review) and. 4.4a (also review) Objectives:

= 9 = x + 8 = = -5x 19. For today: 2.5 (Review) and. 4.4a (also review) Objectives: Math 65 / Notes & Practice #1 / 20 points / Due. / Name: Home Work Practice: Simplify the following expressions by reducing the fractions: 16 = 4 = 8xy =? = 9 40 32 38x 64 16 Solve the following equations

More information

For all questions, answer choice E. NOTA" means none of the above answers is correct.

For all questions, answer choice E. NOTA means none of the above answers is correct. For all questions, answer choice " means none of the above answers is correct. 1. The sum of the integers 1 through n can be modeled by a quadratic polynomial. What is the product of the non-zero coefficients

More information

CC Algebra Quadratic Functions Test Review. 1. The graph of the equation y = x 2 is shown below. 4. Which parabola has an axis of symmetry of x = 1?

CC Algebra Quadratic Functions Test Review. 1. The graph of the equation y = x 2 is shown below. 4. Which parabola has an axis of symmetry of x = 1? Name: CC Algebra Quadratic Functions Test Review Date: 1. The graph of the equation y = x 2 is shown below. 4. Which parabola has an axis of symmetry of x = 1? a. c. c. b. d. Which statement best describes

More information

Algebra 2/Trig Apps: Chapter 5 Quadratics Packet

Algebra 2/Trig Apps: Chapter 5 Quadratics Packet Algebra /Trig Apps: Chapter 5 Quadratics Packet In this unit we will: Determine what the parameters a, h, and k do in the vertex form of a quadratic equation Determine the properties (vertex, axis of symmetry,

More information

Subtract 16 from both sides. Divide both sides by 9. b. Will the swing touch the ground? Explain how you know.

Subtract 16 from both sides. Divide both sides by 9. b. Will the swing touch the ground? Explain how you know. REVIEW EXAMPLES 1) Solve 9x + 16 = 0 for x. 9x + 16 = 0 9x = 16 Original equation. Subtract 16 from both sides. 16 x 9 Divide both sides by 9. 16 x Take the square root of both sides. 9 4 x i 3 Evaluate.

More information

Unit 6: Quadratics. Contents

Unit 6: Quadratics. Contents Unit 6: Quadratics Contents Animated gif Program...6-3 Setting Bounds...6-9 Exploring Quadratic Equations...6-17 Finding Zeros by Factoring...6-3 Finding Zeros Using the Quadratic Formula...6-41 Modeling:

More information

lsolve. 25(x + 3)2-2 = 0

lsolve. 25(x + 3)2-2 = 0 II nrm!: lsolve. 25(x + 3)2-2 = 0 ISolve. 4(x - 7) 2-5 = 0 Isolate the squared term. Move everything but the term being squared to the opposite side of the equal sign. Use opposite operations. Isolate

More information

Secondary Math 2H Unit 3 Notes: Factoring and Solving Quadratics

Secondary Math 2H Unit 3 Notes: Factoring and Solving Quadratics Secondary Math H Unit 3 Notes: Factoring and Solving Quadratics 3.1 Factoring out the Greatest Common Factor (GCF) Factoring: The reverse of multiplying. It means figuring out what you would multiply together

More information

Quadratic Equations and Quadratic Functions

Quadratic Equations and Quadratic Functions Quadratic Equations and Quadratic Functions Andrew Gloag Anne Gloag Say Thanks to the Authors Click http://www.ck1.org/saythanks (No sign in required) To access a customizable version of this book, as

More information

1. Graph (on graph paper) the following equations by creating a table and plotting points on a coordinate grid y = -2x 2 4x + 2 x y.

1. Graph (on graph paper) the following equations by creating a table and plotting points on a coordinate grid y = -2x 2 4x + 2 x y. 1. Graph (on graph paper) the following equations by creating a table and plotting points on a coordinate grid y = -2x 2 4x + 2 x y y = x 2 + 6x -3 x y domain= range= -4-3 -2-1 0 1 2 3 4 domain= range=

More information

Name I.D. Number. Select the response that best completes the statement or answers the question.

Name I.D. Number. Select the response that best completes the statement or answers the question. Name I.D. Number Unit 4 Evaluation Evaluation 04 Second Year Algebra 1 (MTHH 039 059) This evaluation will cover the lessons in this unit. It is open book, meaning you can use your textbook, syllabus,

More information

UNIT 3: MODELING AND ANALYZING QUADRATIC FUNCTIONS

UNIT 3: MODELING AND ANALYZING QUADRATIC FUNCTIONS UNIT 3: MODELING AND ANALYZING QUADRATIC FUNCTIONS This unit investigates quadratic functions. Students study the structure of quadratic expressions and write quadratic expressions in equivalent forms.

More information

Quadratic Functions and Equations

Quadratic Functions and Equations Quadratic Functions and Equations 9A Quadratic Functions 9-1 Quadratic Equations and Functions Lab Explore the Axis of Symmetry 9- Characteristics of Quadratic Functions 9-3 Graphing Quadratic Functions

More information

Chapter 9 Quadratic Functions and Equations

Chapter 9 Quadratic Functions and Equations Chapter 9 Quadratic Functions and Equations 1 9 1Quadratic Graphs and their properties U shaped graph such as the one at the right is called a parabola. A parabola can open upward or downward. A parabola

More information

Applied 30S Unit 1 Quadratic Functions

Applied 30S Unit 1 Quadratic Functions Applied 30S Unit 1 Quadratic Functions Mrs. Kornelsen Teulon Collegiate Institute Learning checklist Quadratics Learning increases when you have a goal to work towards. Use this checklist as guide to track

More information

Algebra I. Polynomials.

Algebra I. Polynomials. 1 Algebra I Polynomials 2015 11 02 www.njctl.org 2 Table of Contents Definitions of Monomials, Polynomials and Degrees Adding and Subtracting Polynomials Multiplying a Polynomial by a Monomial Multiplying

More information

2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY

2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY 2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY The following are topics that you will use in Geometry and should be retained throughout the summer. Please use this practice to review the topics you

More information

Using the Laws of Exponents to Simplify Rational Exponents

Using the Laws of Exponents to Simplify Rational Exponents 6. Explain Radicals and Rational Exponents - Notes Main Ideas/ Questions Essential Question: How do you simplify expressions with rational exponents? Notes/Examples What You Will Learn Evaluate and simplify

More information

Algebra I Polynomials

Algebra I Polynomials Slide 1 / 217 Slide 2 / 217 Algebra I Polynomials 2014-04-24 www.njctl.org Slide 3 / 217 Table of Contents Definitions of Monomials, Polynomials and Degrees Adding and Subtracting Polynomials Multiplying

More information

Ms. Peralta s IM3 HW 5.4. HW 5.4 Solving Quadratic Equations. Solve the following exercises. Use factoring and/or the quadratic formula.

Ms. Peralta s IM3 HW 5.4. HW 5.4 Solving Quadratic Equations. Solve the following exercises. Use factoring and/or the quadratic formula. HW 5.4 HW 5.4 Solving Quadratic Equations Name: Solve the following exercises. Use factoring and/or the quadratic formula. 1. 2. 3. 4. HW 5.4 5. 6. 4x 2 20x + 25 = 36 7. 8. HW 5.4 9. 10. 11. 75x 2 30x

More information

CHAPTER 1 QUADRATIC FUNCTIONS AND FACTORING

CHAPTER 1 QUADRATIC FUNCTIONS AND FACTORING CHAPTER 1 QUADRATIC FUNCTIONS AND FACTORING Big IDEAS: 1) Graphing and writing quadratic functions in several forms ) Solving quadratic equations using a variety of methods 3) Performing operations with

More information

Section 5.4 Quadratic Functions

Section 5.4 Quadratic Functions Math 150 c Lynch 1 of 6 Section 5.4 Quadratic Functions Definition. A quadratic function is one that can be written in the form, f(x) = ax 2 + bx + c, where a, b, and c are real numbers and a 0. This if

More information

Math 2 1. Lesson 4-5: Completing the Square. When a=1 in a perfect square trinomial, then. On your own: a. x 2 18x + = b.

Math 2 1. Lesson 4-5: Completing the Square. When a=1 in a perfect square trinomial, then. On your own: a. x 2 18x + = b. Math 1 Lesson 4-5: Completing the Square Targets: I can identify and complete perfect square trinomials. I can solve quadratic equations by Completing the Square. When a=1 in a perfect square trinomial,

More information

( ) f ( x 1 ) . x 2. To find the average rate of change, use the slope formula, m = f x 2

( ) f ( x 1 ) . x 2. To find the average rate of change, use the slope formula, m = f x 2 Common Core Regents Review Functions Quadratic Functions (Graphs) A quadratic function has the form y = ax 2 + bx + c. It is an equation with a degree of two because its highest exponent is 2. The graph

More information

Math League SCASD. Meet #5. Self-study Packet

Math League SCASD. Meet #5. Self-study Packet Math League SCASD Meet #5 Self-study Packet Problem Categories for this Meet (in addition to topics of earlier meets): 1. Mystery: Problem solving 2. Geometry: Solid Geometry (Volume and Surface Area)

More information

Chapter(5( (Quadratic(Equations( 5.1 Factoring when the Leading Coefficient Equals 1

Chapter(5( (Quadratic(Equations( 5.1 Factoring when the Leading Coefficient Equals 1 .1 Factoring when the Leading Coefficient Equals 1 1... x 6x 8 x 10x + 9 x + 10x + 1 4. (x )( x + 1). (x + 6)(x 4) 6. x(x 6) 7. (x + )(x + ) 8. not factorable 9. (x 6)(x ) 10. (x + 1)(x ) 11. (x + 7)(x

More information

A. B. C. D. Quadratics Practice Test. Question 1. Select the graph of the quadratic function. g (x ) = 1 3 x 2. 3/8/2018 Print Assignment

A. B. C. D. Quadratics Practice Test. Question 1. Select the graph of the quadratic function. g (x ) = 1 3 x 2. 3/8/2018 Print Assignment Question 1. Select the graph of the quadratic function. g (x ) = 1 3 x 2 C. D. https://my.hrw.com/wwtb2/viewer/printall_vs23.html?umk5tfdnj31tcldd29v4nnzkclztk3w8q6wgvr262aca0a5fsymn1tfv8j1vs4qotwclvofjr8xhs0cldd29v4

More information

Solving Equations by Factoring. Solve the quadratic equation x 2 16 by factoring. We write the equation in standard form: x

Solving Equations by Factoring. Solve the quadratic equation x 2 16 by factoring. We write the equation in standard form: x 11.1 E x a m p l e 1 714SECTION 11.1 OBJECTIVES 1. Solve quadratic equations by using the square root method 2. Solve quadratic equations by completing the square Here, we factor the quadratic member of

More information

New Jersey Center for Teaching and Learning. Progressive Mathematics Initiative

New Jersey Center for Teaching and Learning. Progressive Mathematics Initiative Slide 1 / 70 New Jersey Center for Teaching and Learning Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of students

More information

Example: f(x) = 2x² + 1 Solution: Math 2 VM Part 5 Quadratic Functions April 25, 2017

Example: f(x) = 2x² + 1 Solution: Math 2 VM Part 5 Quadratic Functions April 25, 2017 Math 2 Variable Manipulation Part 5 Quadratic Functions MATH 1 REVIEW THE CONCEPT OF FUNCTIONS The concept of a function is both a different way of thinking about equations and a different way of notating

More information

Solving Quadratics Algebraically

Solving Quadratics Algebraically Solving Quadratics Algebraically Table of Contents 1. Introduction to Solving Quadratics. Solving Quadratic Equations using Factoring 3. Solving Quadratic Equations in Context 4. Solving Quadratics using

More information

; Vertex: ( b. 576 feet above the ground?

; Vertex: ( b. 576 feet above the ground? Lesson 8: Applications of Quadratics Quadratic Formula: x = b± b 2 4ac 2a ; Vertex: ( b, f ( b )) 2a 2a Standard: F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand

More information

Quarter 2 400, , , , , , ,000 50,000

Quarter 2 400, , , , , , ,000 50,000 Algebra 2 Quarter 2 Quadratic Functions Introduction to Polynomial Functions Hybrid Electric Vehicles Since 1999, there has been a growing trend in the sales of hybrid electric vehicles. These data show

More information

EX: Simplify the expression. EX: Simplify the expression. EX: Simplify the expression

EX: Simplify the expression. EX: Simplify the expression. EX: Simplify the expression SIMPLIFYING RADICALS EX: Simplify the expression 84x 4 y 3 1.) Start by creating a factor tree for the constant. In this case 84. Keep factoring until all of your nodes are prime. Two factor trees are

More information

Algebra 1: Hutschenreuter Chapter 10 Notes Adding and Subtracting Polynomials

Algebra 1: Hutschenreuter Chapter 10 Notes Adding and Subtracting Polynomials Algebra 1: Hutschenreuter Chapter 10 Notes Name 10.1 Adding and Subtracting Polynomials Polynomial- an expression where terms are being either added and/or subtracted together Ex: 6x 4 + 3x 3 + 5x 2 +

More information

May 16, Aim: To review for Quadratic Function Exam #2 Homework: Study Review Materials. Warm Up - Solve using factoring: 5x 2 + 7x + 2 = 0

May 16, Aim: To review for Quadratic Function Exam #2 Homework: Study Review Materials. Warm Up - Solve using factoring: 5x 2 + 7x + 2 = 0 Aim: To review for Quadratic Function Exam #2 Homework: Study Review Materials Warm Up - Solve using factoring: 5x 2 + 7x + 2 = 0 Review Topic Index 1. Consecutive Integer Word Problems 2. Pythagorean

More information

The x-coordinate of the vertex: The equation of the axis of symmetry:

The x-coordinate of the vertex: The equation of the axis of symmetry: Algebra 2 Notes Section 4.1: Graph Quadratic Functions in Standard Form Objective(s): Vocabulary: I. Quadratic Function: II. Standard Form: III. Parabola: IV. Parent Function for Quadratic Functions: Vertex

More information

Solve for the variable by transforming equations:

Solve for the variable by transforming equations: Cantwell Sacred Heart of Mary High School Math Department Study Guide for the Algebra 1 (or higher) Placement Test Name: Date: School: Solve for the variable by transforming equations: 1. y + 3 = 9. 1

More information

Solving Multi-Step Equations

Solving Multi-Step Equations 1. Clear parentheses using the distributive property. 2. Combine like terms within each side of the equal sign. Solving Multi-Step Equations 3. Add/subtract terms to both sides of the equation to get the

More information

Chapter 8 ~ Quadratic Functions and Equations In this chapter you will study... You can use these skills...

Chapter 8 ~ Quadratic Functions and Equations In this chapter you will study... You can use these skills... Chapter 8 ~ Quadratic Functions and Equations In this chapter you will study... identifying and graphing quadratic functions transforming quadratic equations solving quadratic equations using factoring

More information

Jakarta International School 8 th Grade AG1 Summative Assessment

Jakarta International School 8 th Grade AG1 Summative Assessment Jakarta International School 8 th Grade AG1 Summative Assessment Unit 6: Quadratic Functions Name: Date: Grade: Standard Advanced Highly Advanced Unit 6 Learning Goals NP Green Blue Black Radicals and

More information

PAP Algebra 2. Unit 4B. Quadratics (Part 2) Name Period

PAP Algebra 2. Unit 4B. Quadratics (Part 2) Name Period PAP Algebra Unit 4B Quadratics (Part ) Name Period 1 After Test WS: 4.6 Solve by Factoring PAP Algebra Name Factor. 1. x + 6x + 8. 4x 8x 3 + + 3. x + 7x + 5 4. x 3x 1 + + 5. x + 7x + 6 6. 3x + 10x + 3

More information

Algebra II (Common Core) Summer Assignment Due: September 11, 2017 (First full day of classes) Ms. Vella

Algebra II (Common Core) Summer Assignment Due: September 11, 2017 (First full day of classes) Ms. Vella 1 Algebra II (Common Core) Summer Assignment Due: September 11, 2017 (First full day of classes) Ms. Vella In this summer assignment, you will be reviewing important topics from Algebra I that are crucial

More information

Algebra 2 Honors. Unit 4, Day 1 Period: Date: Graph Quadratic Functions in Standard Form. (Three more problems on the back )

Algebra 2 Honors. Unit 4, Day 1 Period: Date: Graph Quadratic Functions in Standard Form. (Three more problems on the back ) Algebra Honors Name: Unit 4, Day 1 Period: Date: Graph Quadratic Functions in Standard Form 1. y = 3x. y = 5x + 1 3. y = x 5 4. y = 1 5 x 6. y = x + x + 1 7. f(x) = 6x 4x 5 (Three more problems on the

More information

An equation is a statement that states that two expressions are equal. For example:

An equation is a statement that states that two expressions are equal. For example: Section 0.1: Linear Equations Solving linear equation in one variable: An equation is a statement that states that two expressions are equal. For example: (1) 513 (2) 16 (3) 4252 (4) 64153 To solve the

More information

Sect Polynomial and Rational Inequalities

Sect Polynomial and Rational Inequalities 158 Sect 10.2 - Polynomial and Rational Inequalities Concept #1 Solving Inequalities Graphically Definition A Quadratic Inequality is an inequality that can be written in one of the following forms: ax

More information

Park Forest Math Team. Meet #5. Algebra. Self-study Packet

Park Forest Math Team. Meet #5. Algebra. Self-study Packet Park Forest Math Team Meet #5 Self-study Packet Problem Categories for this Meet: 1. Mystery: Problem solving 2. Geometry: Angle measures in plane figures including supplements and complements 3. Number

More information

Common Core Algebra 2. Chapter 3: Quadratic Equations & Complex Numbers

Common Core Algebra 2. Chapter 3: Quadratic Equations & Complex Numbers Common Core Algebra 2 Chapter 3: Quadratic Equations & Complex Numbers 1 Chapter Summary: The strategies presented for solving quadratic equations in this chapter were introduced at the end of Algebra.

More information

8th Grade. Equations with Roots and Radicals.

8th Grade. Equations with Roots and Radicals. 1 8th Grade Equations with Roots and Radicals 2015 12 17 www.njctl.org 2 Table of Contents Radical Expressions Containing Variables Click on topic to go to that section. Simplifying Non Perfect Square

More information

Park Forest Math Team. Meet #5. Self-study Packet

Park Forest Math Team. Meet #5. Self-study Packet Park Forest Math Team Meet #5 Self-study Packet Problem Categories for this Meet (in addition to topics of earlier meets): 1. Mystery: Problem solving 2. Geometry: Solid Geometry (Volume and Surface Area)

More information

UNIT 2B QUADRATICS II

UNIT 2B QUADRATICS II UNIT 2B QUADRATICS II M2 12.1-8, M2 12.10, M1 4.4 2B.1 Quadratic Graphs Objective I will be able to identify quadratic functions and their vertices, graph them and adjust the height and width of the parabolas.

More information

Algebra 1. Math Review Packet. Equations, Inequalities, Linear Functions, Linear Systems, Exponents, Polynomials, Factoring, Quadratics, Radicals

Algebra 1. Math Review Packet. Equations, Inequalities, Linear Functions, Linear Systems, Exponents, Polynomials, Factoring, Quadratics, Radicals Algebra 1 Math Review Packet Equations, Inequalities, Linear Functions, Linear Systems, Exponents, Polynomials, Factoring, Quadratics, Radicals 2017 Math in the Middle 1. Clear parentheses using the distributive

More information

6.4. The Quadratic Formula. LEARN ABOUT the Math. Selecting a strategy to solve a quadratic equation. 2x 2 + 4x - 10 = 0

6.4. The Quadratic Formula. LEARN ABOUT the Math. Selecting a strategy to solve a quadratic equation. 2x 2 + 4x - 10 = 0 6.4 The Quadratic Formula YOU WILL NEED graphing calculator GOAL Understand the development of the quadratic formula, and use the quadratic formula to solve quadratic equations. LEARN ABOUT the Math Devlin

More information

Study Guide for Math 095

Study Guide for Math 095 Study Guide for Math 095 David G. Radcliffe November 7, 1994 1 The Real Number System Writing a fraction in lowest terms. 1. Find the largest number that will divide into both the numerator and the denominator.

More information

9.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED LESSON

9.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED LESSON CONDENSED LESSON 9.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations solve

More information

Solving Linear-Quadratic Systems

Solving Linear-Quadratic Systems 36 LESSON Solving Linear-Quadratic Sstems UNDERSTAND A sstem of two or more equations can include linear and nonlinear equations. In a linear-quadratic sstem, there is one linear equation and one quadratic

More information

Section 5: Quadratic Equations and Functions Part 1

Section 5: Quadratic Equations and Functions Part 1 Section 5: Quadratic Equations and Functions Part 1 Topic 1: Real-World Examples of Quadratic Functions... 121 Topic 2: Factoring Quadratic Expressions... 125 Topic 3: Solving Quadratic Equations by Factoring...

More information

Quadratics Unit Review

Quadratics Unit Review Name: Class: Date: Quadratics Unit Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. (2 points) Consider the graph of the equation y = ax 2 + bx +

More information

Unit 3. Expressions and Equations. 118 Jordan School District

Unit 3. Expressions and Equations. 118 Jordan School District Unit 3 Epressions and Equations 118 Unit 3 Cluster 1 (A.SSE.): Interpret the Structure of Epressions Cluster 1: Interpret the structure of epressions 3.1. Recognize functions that are quadratic in nature

More information

Looking Ahead to Chapter 10

Looking Ahead to Chapter 10 Looking Ahead to Chapter Focus In Chapter, you will learn about polynomials, including how to add, subtract, multiply, and divide polynomials. You will also learn about polynomial and rational functions.

More information

Roots are: Solving Quadratics. Graph: y = 2x 2 2 y = x 2 x 12 y = x 2 + 6x + 9 y = x 2 + 6x + 3. real, rational. real, rational. real, rational, equal

Roots are: Solving Quadratics. Graph: y = 2x 2 2 y = x 2 x 12 y = x 2 + 6x + 9 y = x 2 + 6x + 3. real, rational. real, rational. real, rational, equal Solving Quadratics Graph: y = 2x 2 2 y = x 2 x 12 y = x 2 + 6x + 9 y = x 2 + 6x + 3 Roots are: real, rational real, rational real, rational, equal real, irrational 1 To find the roots algebraically, make

More information

Section 3.1 Quadratic Functions

Section 3.1 Quadratic Functions Chapter 3 Lecture Notes Page 1 of 72 Section 3.1 Quadratic Functions Objectives: Compare two different forms of writing a quadratic function Find the equation of a quadratic function (given points) Application

More information

QUADRATIC FUNCTIONS AND MODELS

QUADRATIC FUNCTIONS AND MODELS QUADRATIC FUNCTIONS AND MODELS What You Should Learn Analyze graphs of quadratic functions. Write quadratic functions in standard form and use the results to sketch graphs of functions. Find minimum and

More information

Unit 2 Quadratics. Mrs. Valentine Math 3

Unit 2 Quadratics. Mrs. Valentine Math 3 Unit 2 Quadratics Mrs. Valentine Math 3 2.1 Factoring and the Quadratic Formula Factoring ax 2 + bx + c when a = ±1 Reverse FOIL method Find factors of c that add up to b. Using the factors, write the

More information

AP PHYSICS SUMMER ASSIGNMENT

AP PHYSICS SUMMER ASSIGNMENT AP PHYSICS SUMMER ASSIGNMENT There are two parts of the summer assignment, both parts mirror the course. The first part is problem solving, where there are 14 math problems that you are given to solve

More information

. State the important connection between the coefficients of the given trinomials and the values you found for r.

. State the important connection between the coefficients of the given trinomials and the values you found for r. Motivational Problems on Quadratics 1 1. Factor the following perfect-square trinomials : (a) x 1x 36 (b) x 14x 49 (c) x 0x 100 As suggested, these should all look like either ( x r) or ( x r). State the

More information

Chapter 1 Notes: Quadratic Functions

Chapter 1 Notes: Quadratic Functions 19 Chapter 1 Notes: Quadratic Functions (Textbook Lessons 1.1 1.2) Graphing Quadratic Function A function defined by an equation of the form, The graph is a U-shape called a. Standard Form Vertex Form

More information

Completing the Square

Completing the Square 5-7 Completing the Square TEKS FOCUS TEKS (4)(F) Solve quadratic and square root equations. TEKS (1)(A) Apply mathematics to problems arising in everyday life, society, and the workplace. Additional TEKS

More information

The Quadratic Formula. ax 2 bx c 0 where a 0. Deriving the Quadratic Formula. Isolate the constant on the right side of the equation.

The Quadratic Formula. ax 2 bx c 0 where a 0. Deriving the Quadratic Formula. Isolate the constant on the right side of the equation. SECTION 11.2 11.2 The Quadratic Formula 11.2 OBJECTIVES 1. Solve quadratic equations by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation by using the discriminant

More information

test corrections graphing calculator factoring test

test corrections graphing calculator factoring test Warm-Up 1. Please turn in your test corrections to the inbox 2. You need your graphing calculator for today s lesson 3. If you need to take your factoring test, please come talk to Ms. Barger before class

More information

Test 4 also includes review problems from earlier sections so study test reviews 1, 2, and 3 also.

Test 4 also includes review problems from earlier sections so study test reviews 1, 2, and 3 also. MATD 0370 ELEMENTARY ALGEBRA REVIEW FOR TEST 4 (1.1-10.1, not including 8.2) Test 4 also includes review problems from earlier sections so study test reviews 1, 2, and 3 also. 1. Factor completely: a 2

More information

3.1. QUADRATIC FUNCTIONS AND MODELS

3.1. QUADRATIC FUNCTIONS AND MODELS 3.1. QUADRATIC FUNCTIONS AND MODELS 1 What You Should Learn Analyze graphs of quadratic functions. Write quadratic functions in standard form and use the results to sketch graphs of functions. Find minimum

More information

Section 5.0A Factoring Part 1

Section 5.0A Factoring Part 1 Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)(3 + 5)

More information

3.1 Solving Quadratic Equations by Factoring

3.1 Solving Quadratic Equations by Factoring 3.1 Solving Quadratic Equations by Factoring A function of degree (meaning the highest exponent on the variable is ) is called a Quadratic Function. Quadratic functions are written as, for example, f(x)

More information

Algebra II Honors Unit 3 Assessment Review Quadratic Functions. Formula Box. f ( x) 2 x 3 25 from the parent graph of

Algebra II Honors Unit 3 Assessment Review Quadratic Functions. Formula Box. f ( x) 2 x 3 25 from the parent graph of Name: Algebra II Honors Unit 3 Assessment Review Quadratic Functions Date: Formula Box x = b a x = b ± b 4ac a h 6t h 0 ) What are the solutions of x 3 5? x 8or x ) Describe the transformation of f ( x)

More information

9.4 Start Thinking. 9.4 Warm Up. 9.4 Cumulative Review Warm Up. Use a graphing calculator to graph ( )

9.4 Start Thinking. 9.4 Warm Up. 9.4 Cumulative Review Warm Up. Use a graphing calculator to graph ( ) 9.4 Start Thinking Use a graphing calculator to graph ( ) f x = x + 4x 1. Find the minimum of the function using the CALC feature on the graphing calculator. Explain the relationship between the minimum

More information

MATH 121: EXTRA PRACTICE FOR TEST 2. Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam.

MATH 121: EXTRA PRACTICE FOR TEST 2. Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam. MATH 121: EXTRA PRACTICE FOR TEST 2 Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam. 1 Linear Functions 1. Consider the functions f(x) = 3x + 5 and g(x)

More information

5.4 - Quadratic Functions

5.4 - Quadratic Functions Fry TAMU Spring 2017 Math 150 Notes Section 5.4 Page! 92 5.4 - Quadratic Functions Definition: A function is one that can be written in the form f (x) = where a, b, and c are real numbers and a 0. (What

More information