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.

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1 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, then 1 b c. On your own: a. x 18x + = b. x + x + = c. x + 1x + =

2 Math Rewriting a Quadratic Equation from standard form to vertex form when a = 1 Here is an example: Rewrite the following quadratic in vertex form: y = x - 16x + 5 Step 1 Subtract c from both sides Step Take b and add to both sides y = x - 16x + 64 Step 3 Factor the perfect square trinomial y + 59 = (x - 8) The vertex is (8,-59) y = (x 8) y = x + 4x y = x + 5x + 34 **Could you do #3 using formula instead of completing the square?

3 Math 3 Homework 4-5: Completing the Square (a = 1) Write each quadratic function in vertex form by completing the square. 1) y = x 14x + 54 ) y = x + 8x + 7 3) y = x + 10x + 8 4) y = x 4x 5) y = x 1x 41 6) y = x 8x ) y = x 4x 8) y = x + 8x + 18

4 Math 4 Lesson 4-6: Solving Equations by Completing the Square Targets: I can solve quadratic equations by Completing the Square You can also solve equations by COMPLETING THE SQUARE. The reason you can do this is because you can square root a binomial squared. In other words: ( a b) a b Be careful when you square root binomials. Find the mistake in this statement. Error Analysis: x 4 x Solve these equations by completing the square. 1. x + 9x + 15 = 0. m + 16m = t - 4t -165 = 0 4. m + 7m - 94=0

5 Math 5 Homework 4-6: Solving Quadratics by Completing the Square Solve each quadratic equation by completing the square and taking the square root. 1) x + 6x 7 = 0 ) x + 6x + 10 = 0 3) x 4x 8 = 0 4) x 48 = 0 5) x x 1 = 0 6) x 8x + 4 = 0 7) 5x 6x = 8 8) x 4x 165 = 0

6 Math 6 Lesson 4-7: Introduction to the Quadratic Formula Targets: I can solve quadratics by using the quadratic formula. In order to use the quadratic formula to solve a quadratic: you must first set the quadratic equal to 0.

7 Math 7 8.) A batter hits a baseball. The equation y= -.005x + 0.7x models the path of the ball, where x is the horizontal distance, in feet, the ball travels and y is the height, in feet, of the ball. How far does the ball land from the batter. Round to nearest tenth of a foot. What was the maximum height of the ball? How far from the batter was the ball when it hit the maximum height? How far from the batter was the ball when the ball was 15 feet high? 9.) Pete hit a fly ball whose path can be described by the function (x) = x +. 1x In this function, x is the distance on the ground (in feet) of the ball from home plate and h(x) is the height (in feet) of the ball. The ball is traveling toward the outfield fence, which is 7 feet high and 389 feet from home plate. Could the ball go over the fence? Explain your reasoning. 10.) Consider the equation y = x + 4x + 1. Write a brief description of the graph of the equation. (You do not have to draw the graph.) Your description should include facts such as the shape of the graph, the number of x-intercepts, the y-intercept, the vertex, and the equation of the axis of symmetry.

8 Math 8 Homework 4-7 Quadratic Formula Use the quadratic formula to find the zeros of each function. Approximate the zeros to the nearest tenth 1) y = x x 3 5) 5 = 3x 4x Zeros:, Zeros:, ) y = x 4x 3 6) y = x 16x + 8 Zeros:, Zeros:, 3) y = x 8x ) 6 = x Zeros:, Zeros:, 4) y = x 16x 3 8) 1 = x + 6x Zeros:, Zeros:,

9 Math 9 Lesson 4-9: Solving Quadratics Targets: I can solve quadratics by factoring, using the quadratic formula, using my graphing calculator, and completing the square. Some quadratic equations can be solved by using inverse operations. This can be done when b=0. Remember that when you square root both sides that negative values for x. x x, so you will need to give both the positive and 1. x 16 = 0. 3x + 6 = 0 3. (x - 7) = 0 To solve quadratics where b 0 it is slightly more complicated, here are some important things to remember. to quadratics are also called,, and. A is defined as the x value for which f(x) = 0. We have several different methods that we can use to solve quadratics but setting the equation equal to zero is always our first step. This list is not exhaustive but in this class this is how we will solve quadratics. Factoring Quadratic Formula Technology Completing the Square Zero Product Property: If ab = 0, then either a or b must equal zero Solve by factoring. You will need to use the zero product property to find solutions. Let's practice the zero product property first: 1.) (x - )(x+9)=0.) (x+3)(x-4) =0 3.) (7n-)(5n-4)=0 4.) x(x-3) = 0

10 Math 10 Solve by factoring. 5.) 0 x 10x 4 6.) 0 y y 7.) 6n n 40 8.) 5x -10x = 0 9.) c = 5c Can you graph by factoring. Since you can find the solution to a quadratic equation by factoring, the x intercepts are the same as the solutions. Using symmetry, you can find the axis of symmetry. Graph: Factor to find the zeros and use symmetry to find the vertex. 10. y x 9x 18 Explain how to find the axis of symmetry using the roots.

11 Math 11 Solve using quadratic formula. Round answers to nearest hundredth if necessary. 11.) 10z 13z ) 1 n 11n 13.) m m 7 14.) 3x 11x ) 7x x 8 Go back and do problems using your calculator to find your solutions. Make sure you use your calc function and don't just use trace. Trace does not give as good of an approximation. Graph these quadratics using their zeros and vertices. 16.) y = 6x + 13x ) y = 8x + 14x - 15

12 Math 1 Homework 4-9: Solving Quadratics 1. x = 18 9x. 7x 4 = 35x 3. x + x 1 = 0 4. x = x 5x 14 = 0 6. x + 8 = x 3x 5 = x + 6x 16 = 5x

13 Math x + 8x + 7 = x = (x 6) = x + 7 = 33k 13. x + x = x + 10 = 91 * x 8x 7. = 0 * x =.6x Solve with graphing calculator. Round your answer to the nearest hundredth.

14 Math 14 Lesson 4-10: Applications with Quadratics Gravitational constant (g) = 9.8m / s or 3 ft / s 1 The height of a projectile given as a function of time is given by the formula: h gt v0t h0 where v0 is the initial velocity and h 0 is the initial height. If an object is dropped, the height can be given by the 1 formula h gt h0 1.) An acorn drops form a tree branch 70 ft above the ground. The function h= -16t +70 gives the height h of the acorn (in feet) after t seconds. What is the graph of this function. At what time does the acorn hit the ground? a.) What is a reasonable domain for this function? b.) When is the vertex of a parabola a minimum? c.) When is the vertex of a parabola a maximum?.) Suppose a person is riding in a hot- air balloon, 154 ft above the ground. He drops an apple. The height, h, in feet, of the apple above the ground is given by the formula h = -16t +154, where t is time in seconds. a.) What time does the apple hit the ground? b.) How far has to apple fallen from t=0 to t=1? c.) Does the ball fall the same distance from t=1 to t= as it does from t=0 to t=1? Explain

15 Math 15 3.) A blueprint for a 15ft-by-9ft rectangular wall has a square window in the center. If each side of the window is x ft, the function y=135 - x gives the area (in sq. ft)of the wall minus the area of the window. a.) Graph the function. b.) What is a reasonable domain of the function? Explain. c.) What is the range of the function? Explain. d.) Estimate the side length of the window if the area of the wall is 117 ft. 4.) A baseball is thrown into the air with an upward velocity of 30 ft/s. Write a function that gives the height 1 h, in feet, after t seconds of the baseball. Remember ( h gt v0t h0 ) b.) When will the baseball reach its maximum height? c.) What is the maximum height of the ball? d.) When will the ball hit the ground? e.) When will the ball hit a height of 1 feet?

16 Math 16 5.) Suppose you have 100 ft of string to rope of a rectangle section for a bake sale at a school fair. The function A= -x +50x give the area of the section in sq. ft, where x is the width in feet. What width gives you the maximum area you can rope off? b.) What is the maximum area? c.) What is the range of the function? 6.) What translation maps y= x onto f(x)= x + x - 3? 6. 7.) 8.) You have enough shrubs to cover an area of 100 ft. What is the radius of the largest circular region you can plant with these shrubs?

17 Math 17 9.) Suppose a ball is thrown upward from a height of 5 feet with an initial velocity of 35 ft/sec. a.) Write an equation relating the time t and the height h of the ball. b.) Find the height of the ball after seconds. c.) Is the ball still in the air after 3 seconds? Explain. d.) When did the ball reach its max height? e.) How high was the ball when it reached its maximum height? 10.) For what values of n will the equation x =n have two solutions? 10. b.) What values of n will have one solution? c.) What values of n will have no solution? b. c. 11.) The trapezoid has an area of 1960 cm. Use 1 the formula A h( b 1 b ), where A represents the area of the trapezoid, he represents its height, and b andb 1 represent its bases, to find the value of y. 11.

18 Math 18 1.) You have a rectangular koi pond that measures 6ft by 8ft. You have enough concrete to cover 7 ft for a walkway, as shown in the diagram. Write a function for find the area of the concrete walkway. b.) What is the maximum width of the walkway? 13.) Find the equation of a quadratic congruent to y = x that has roots of and -5. Give the equation in standard form. 13.