Secondary Math 2 Honors Unit 4 Graphing Quadratic Functions
|
|
- Gillian Francis
- 5 years ago
- Views:
Transcription
1 SMH Secondary Math Honors Unit 4 Graphing Quadratic Functions 4.0 Forms of Quadratic Functions Form: ( ) f = a + b + c, where a 0. There are no parentheses. f = Eample: ( ) Form: f ( ) = a( p)( q), where a 0. Written as a multiplication problem. Also known as intercept form. f = Eample: ( ) ( )( ) Form: f ( ) = a( h) + k, where a 0. only shows up once, as part of a perfect square. Eample: ( ) ( ) f = Conic Form of a parabola: 4pp(yy kk) = ( h) or 4pp( h) = (yy kk) Eamples: 4(yy ) = ( + 5) or 8( + 6) = (yy 1) Eamples: State whether each quadratic function is in standard, factored, or verte form. f = a) f ( ) = ( + 3)( 5) b) ( ) ( 4) f = + 5 c) ( ) d) f ( ) = + 5 e) f ( ) = 3( ) f) f ( ) ( ) = g) f ( ) = ( + 5) h) f ( ) = i) f ( ) = 5
2 Vocabulary: 4.1 Graphing Quadratic Functions: Verte and Ais of Symmetry : The shape of the graph of a quadratic function. : A line that cuts a parabola in half. If you were to fold a parabola along its ais of symmetry, the two sides would overlap. The equation of the ais of symmetry looks like = #. : The tip of the parabola the point at which it changes direction. If the parabola opens up ( a > 0 ), the verte is the lowest point on the graph, or the minimum point. If the parabola opens down ( a < 0 ), the verte is the highest point on the graph, or the maimum point. a > 0 a < 0 Finding the 1. Plug in 0 for.. Simplify. Don t forget order of operations. Verte Form of a Quadratic Function: y = a( h) + k ( hk, ) = h The sign of h is the opposite of the sign in the equation. h moves the graph of y = right and left in the opposite direction as the sign in the equation (but the same direction as the sign of h itself). The sign of k is the same as the sign in the equation. k moves the graph of y = up and down in the same direction as the sign in the equation. o For y = ( ) + 5, h and k 5.,5 and the ais of symmetry is =. The graph of y o For y ( ) = = The verte is ( ) = moved right and up 5. = + 3 7, h 3 and k 7. = 3. The graph of = = The verte is ( 3, 7) y = moved left 3 and down 7. and the ais of symmetry is
3 Opens up if a is positive. Opens down if a is negative. Eamples: For each function, do the following: 1) State the coordinates of the verte. ) State the direction of the opening, that is, whether the parabola opens up or down. 3) Find the y-intercept. 4) Draw a rough sketch of the graph. 5) Find the Domain and Range a) ( ) y = Direction: ais of symmetry: sketch graph: Domain: Range: f = Direction: ais of symmetry: sketch graph: c) ( ) ( ) Domain: Range: Vertical Stretch: a changes how wide or narrow the graph is. o If a > 1, the graph is narrower than the graph of o If a < 1, the graph is wider than the graph of y =. y =. Figure out the eact shape of the graph by making an,y table. Always use the verte as one point. Then choose two -values on each side of the verte to plug into the equation to find the corresponding y-coordinates. A shortcut is to use counting patterns to graph the parabola. Start at the verte, then count: 1, a, 4a If a is negative, count down instead of up. 3, 9a, etc. o For y = ( 3) 4, a =. Start at the verte ( 3, 4 ), and count 1, ;, 8; 3, 18
4 Eamples: Fill in the requested information for each function. Then draw the graph. a) ( ) ( ) f = 1 4 a = h= k = Domain: Range: b) ff() = ( + ) 1 a = h= k = Domain: Range: Standard Form: f ( ) = a + b + c Just like with the other forms, the graph opens up if a is positive and opens down if a is negative. b o The -coordinate of the verte is. (The opposite of b divided by times a) a o To find the y-coordinate, plug the -coordinate into the original equation.
5 Eamples: Fill in the requested information for each function. Then draw the graph. f = a) ( ) a = b= c= What is the maimum/minimum value? Domain: Range: b) y 4 = + a= b= c= What is the maimum/minimum value? Factored Form: ff() = aa( pp)(yy qq) Like other forms, a is the vertical stretch (pp, 0) and (qq, 0) are the -intercepts (zeroes). The -value of the verte is eactly half-way between them Evaluate the function at = pp+qq to find the y value of the verte.
6 Eamples: Fill in the requested information for each function. Then draw the graph. a) ff() = ( 5)( + 1) pp = qq = direction of opening: verte: Is it ma or min: Domain: Range: b) yy = 1 ( + 3)( 1) pp = qq = direction of opening: verte: Is it ma or min: Domain: Range:
7 4. Graphing using zeroes, solutions, roots, and -intercepts of a Function: The values of that make f ( ) or y equal zero. If the zeros are real, they tell you the places where the graph crosses the -ais, or the -intercepts of the graph. Other words for zeros: solutions to f ( ) = 0, roots, -intercepts. Finding zeros (-intercepts): 1. Change y or f ( ) to 0.. Solve for. If the equation is in factored form, solving for is easy just think What would have to be to make each set of parentheses equal to 0? If the equation is in standard form, solve by factoring or by using quadratic formula If the equation is in verte form, get the perfect square by itself, take the square root of both sides (don t forget the ±), then solve for. If your answers are imaginary (negative under the square root), the graph doesn t have -intercepts. Eamples: For each equation, find the zeros and state whether the graph opens up or down. Then match the equation to the correct graph. a) y = ( 1)( + 3) b) f ( ) = 1 ( + ) c) y 3( 3)( 1) f = = d) ( ) ( ) Graph 1: Graph : Graph 3: Graph 4:
8 Eamples: For each function, do the following: 1) State whether the parabola has a maimum or minimum. ) State whether the parabola opens up or down. 3) Find the -intercept(s). 4) Find the y-intercept. 5) Draw a rough sketch of the graph. a) f ( ) ( 4)( 1) = + Ma/Min: Direction: -intercept(s): sketch graph: Graphing from Factored Form: 1. Determine whether the parabola will open up or open down.. Find the zeros or -intercepts. Let f( ) = 0 and solve the equation. Mark the -intercepts on the graph. 3. Find the y-intercept but substituting in = 0. Mark this point on the graph. 4. Find the ais of symmetry and the verte. Use the method from 4.1 If there is only one zero, then the ais of symmetry will run vertically through that point and that -intercept will also be the verte. 5. Use the pattern from Section 4.1 to find other points on the graph now that you have the location of the verte. **Remember that a parabola is a smooth curve. Do not draw straight lines! Eamples: Fill in the requested information for each function. Then draw the graph. f = + a) ( ) ( 1)( 3) Zeros (-intercepts): Domain: Range:
9 f = b) ( ) Factored Form: f ( ) = Zeros (-intercepts): f = + 4 c) ( ) Factored Form: f ( ) = Zeros (-intercepts):
10 f = d) ( ) Factored Form: f ( ) = Zeros (-intercepts): e) ff() = Factored Form: f ( ) = Zeros (-intercepts):
11 4.3 Writing equations from a graph or from set of information Eamples: Write a quadratic equation or function in Verte Form: f( ) = y= a ( h) + k If you know the verte of a parabola, ( hk, ), then you still need at least one other point on the parabola in order to write an equation. Use the verte form and fill in all the information you have. Then use the point on the parabola and substitute in for and y. Solve for a. Write your final equation a) (,1 ), passes through ( 4,13 ) b) ( 5,3), passes through ( 1, 9) Eamples: Write the equation of each parabola based on the information in the graph. Follow the steps outlined above. Leave the equations in Verte Form. a) y 10 y
12 Eamples: Write a quadratic equation or function in Factored Form: f( ) = y= a ( p) ( q) Use the factored form if you know the roots (a.k.a. solutions, -intercepts, zeros). You will still need to know at least one other point on the parabola in order to write an equation. Use the factored form and fill in all the information you have. Then use the point on the parabola and substitute in for and y. Solve for a. Write your final equation a) Roots: ( 3,0 ) &(,0), goes through (, 4) b) -intercept: (3, 0) & (, 0), goes through (0, 1) = = goes through ( 6, 10) c) Zeros: 1 & 3, d) Roots: = 7 & = 7, goes through ( 6, 9) e) Solutions: 8 i & 8 i, = = passes through (, 04)
13 Eamples: Write the equation of each parabola based on the information in the graph. Leave the equations in Factored Form. y 10 a) b) y Eamples: Write a quadratic equation or function in Standard Form: ff() = aa + bbbb + cc First write the equation in either Verte Form or Factored Form (whichever seems easier) Then use correct order of operations to multiply/distribute in order to get rid of parenthesis List the three terms in correct order:, then, then the constant term a) Use the graph just above to write the equation in Standard Form b) Write one of the verte form equations in Standard Form c) Write the equation with the following characteristics in Factored Form, Verte Form and Standard Form (yes, we ll need to complete the square to get it into verte form ) roots at ( 3, 0) and (5, 0) and goes through the point (, 15)
14 4.4 Solving Quadratic Inequalities & Systems of Equations by Graphing Eamples: Solve each inequality using the graph of ( ) f = + 3. Notice that each of these inequalities involves the value of + 3, which is represented by the y-coordinate of the graph. In each case, we are trying to figure out what -values (-coordinates) make the inequality true. When trying to find where + 3 > 0, we are trying to figure out what -coordinates have a y-coordinate that is bigger than zero in other words, where is the graph above the -ais? f = + 3 a) + 3> 0 b) ( ) c) + 3< 0 d) Solving a Quadratic Inequality Using the Graph: 1. Write the inequality in standard form. Replace the inequality sign with an equal sign and solve the equation a + b + c = 0 by factoring, completing the square, or using the quadratic formula. This gives you the -intercepts of the graph of y = a + b + c.. Graph y = a + b + c. The graph does not have to be very detailed. A rough sketch of a parabola opening in the correct direction with the correct -intercepts is all you need. 3. The solutions of The solutions of The solutions of The solutions of a + b + c > 0 are the -values for which the graph is above the -ais. a + b + c 0 are the -values for which the graph is on or above the -ais. a + b + c < 0 are the -values for which the graph is below the -ais. a + b + c 0 are the -values for which the graph is on or below the -ais. 4. If the inequality involves or, the -intercepts are included in the solution set (use brackets). If the inequality involves < or >, the -intercepts are not included in the solution set (use parentheses).
15 Eamples: Solve each quadratic inequality and write the solution set in interval notation. a) ( 3)( + 1) 0 b) ( )( ) 7 5 < 0 c) + 5> 0 d) e) < f)
16 g) h) > + 6 Solving Systems of Equations by Graphing Solving a system of equations means finding the values of and y that make both equations true. The solutions are usually written as ordered pairs ( y, ). Solving by graphing: 1. Solve both equations for y.. Graph both equations using y = m + b, transformations, or, y tables. 3. The points where the two graphs intersect (cross) are the solutions. 4. Write the solutions as ordered pairs. 5. Eamples: Solve by graphing. + y = 6 a) y = + 4 b) 6 y = 4 6 3y = 3
Graphs and Solutions for Quadratic Equations
Format y = a + b + c where a 0 Graphs and Solutions for Quadratic Equations Graphing a quadratic equation creates a parabola. If a is positive, the parabola opens up or is called a smiley face. If a is
More informationName: Period: SM Starter on Reading Quadratic Graph. This graph and equation represent the path of an object being thrown.
SM Name: Period: 7.5 Starter on Reading Quadratic Graph This graph and equation represent the path of an object being thrown. 1. What is the -ais measuring?. What is the y-ais measuring? 3. What are the
More informationLearning Targets: Standard Form: Quadratic Function. Parabola. Vertex Max/Min. x-coordinate of vertex Axis of symmetry. y-intercept.
Name: Hour: Algebra A Lesson:.1 Graphing Quadratic Functions Learning Targets: Term Picture/Formula In your own words: Quadratic Function Standard Form: Parabola Verte Ma/Min -coordinate of verte Ais of
More informationAlgebra II Midterm Exam Review Packet
Algebra II Midterm Eam Review Packet Name: Hour: CHAPTER 1 Midterm Review Evaluate the power. 1.. 5 5. 6. 7 Find the value of each epression given the value of each variable. 5. 10 when 5 10 6. when 6
More informationACCUPLACER MATH 0311 OR MATH 0120
The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 0 OR MATH 00 http://www.academics.utep.edu/tlc MATH 0 OR MATH 00 Page Factoring Factoring Eercises 8 Factoring Answer to Eercises
More informationCHAPTER 3 : QUADRARIC FUNCTIONS MODULE CONCEPT MAP Eercise 1 3. Recognizing the quadratic functions Graphs of quadratic functions 4 Eercis
ADDITIONAL MATHEMATICS MODULE 5 QUADRATIC FUNCTIONS CHAPTER 3 : QUADRARIC FUNCTIONS MODULE 5 3.1 CONCEPT MAP Eercise 1 3. Recognizing the quadratic functions 3 3.3 Graphs of quadratic functions 4 Eercise
More informationOne Solution Two Solutions Three Solutions Four Solutions. Since both equations equal y we can set them equal Combine like terms Factor Solve for x
Algebra Notes Quadratic Systems Name: Block: Date: Last class we discussed linear systems. The only possibilities we had we 1 solution, no solution or infinite solutions. With quadratic systems we have
More informationVertex. March 23, Ch 9 Guided Notes.notebook
March, 07 9 Quadratic Graphs and Their Properties A quadratic function is a function that can be written in the form: Verte Its graph looks like... which we call a parabola. The simplest quadratic function
More informationUnit 8 - Polynomial and Rational Functions Classwork
Unit 8 - Polynomial and Rational Functions Classwork This unit begins with a study of polynomial functions. Polynomials are in the form: f ( x) = a n x n + a n 1 x n 1 + a n 2 x n 2 +... + a 2 x 2 + a
More informationVertex Form of a Parabola
Verte Form of a Parabola In this investigation ou will graph different parabolas and compare them to what is known as the Basic Parabola. THE BASIC PARABOLA Equation = 2-3 -2-1 0 1 2 3 verte? What s the
More information8 f(8) = 0 (8,0) 4 f(4) = 4 (4, 4) 2 f(2) = 3 (2, 3) 6 f(6) = 3 (6, 3) Outputs. Inputs
In the previous set of notes we covered how to transform a graph by stretching or compressing it vertically. In this lesson we will focus on stretching or compressing a graph horizontally, which like the
More informationUnit 2 Notes Packet on Quadratic Functions and Factoring
Name: Period: Unit Notes Packet on Quadratic Functions and Factoring Notes #: Graphing quadratic equations in standard form, verte form, and intercept form. A. Intro to Graphs of Quadratic Equations: a
More informationModule 2, Section 2 Solving Equations
Principles of Mathematics Section, Introduction 03 Introduction Module, Section Solving Equations In this section, you will learn to solve quadratic equations graphically, by factoring, and by applying
More informationSection 7.1 Objective 1: Solve Quadratic Equations Using the Square Root Property Video Length 12:12
Section 7.1 Video Guide Solving Quadratic Equations by Completing the Square Objectives: 1. Solve Quadratic Equations Using the Square Root Property. Complete the Square in One Variable 3. Solve Quadratic
More informationAlgebra 2 Unit 9 (Chapter 9)
Algebra Unit 9 (Chapter 9) 0. Spiral Review Worksheet 0. Find verte, line of symmetry, focus and directri of a parabola. (Section 9.) Worksheet 5. Find the center and radius of a circle. (Section 9.3)
More informationKEY Algebra: Unit 9 Quadratic Functions and Relations Class Notes 10-1
Name: KEY Date: Algebra: Unit 9 Quadratic Functions and Relations Class Notes 10-1 Anatomy of a parabola: 1. Use the graph of y 6 5shown below to identify each of the following: y 4 identify each of the
More informationLesson #33 Solving Incomplete Quadratics
Lesson # Solving Incomplete Quadratics A.A.4 Know and apply the technique of completing the square ~ 1 ~ We can also set up any quadratic to solve it in this way by completing the square, the technique
More informationQ.3 Properties and Graphs of Quadratic Functions
374 Q.3 Properties and Graphs of Quadratic Functions In this section, we explore an alternative way of graphing quadratic functions. It turns out that if a quadratic function is given in form, ff() = aa(
More informationLesson Goals. Unit 4 Polynomial/Rational Functions Quadratic Functions (Chap 0.3) Family of Quadratic Functions. Parabolas
Unit 4 Polnomial/Rational Functions Quadratic Functions (Chap 0.3) William (Bill) Finch Lesson Goals When ou have completed this lesson ou will: Graph and analze the graphs of quadratic functions. Solve
More information1 a) Remember, the negative in the front and the negative in the exponent have nothing to do w/ 1 each other. Answer: 3/ 2 3/ 4. 8x y.
AP Calculus Summer Packer Key a) Remember, the negative in the front and the negative in the eponent have nothing to do w/ each other. Answer: b) Answer: c) Answer: ( ) 4 5 = 5 or 0 /. 9 8 d) The 6,, and
More informationQUADRATIC GRAPHS ALGEBRA 2. Dr Adrian Jannetta MIMA CMath FRAS INU0114/514 (MATHS 1) Quadratic Graphs 1/ 16 Adrian Jannetta
QUADRATIC GRAPHS ALGEBRA 2 INU0114/514 (MATHS 1) Dr Adrian Jannetta MIMA CMath FRAS Quadratic Graphs 1/ 16 Adrian Jannetta Objectives Be able to sketch the graph of a quadratic function Recognise the shape
More informationRoots 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 information4-1 Graphing Quadratic Functions
4-1 Graphing Quadratic Functions Quadratic Function in standard form: f() a b c The graph of a quadratic function is a. y intercept Ais of symmetry -coordinate of verte coordinate of verte 1) f ( ) 4 a=
More informationCOUNCIL ROCK HIGH SCHOOL MATHEMATICS. A Note Guideline of Algebraic Concepts. Designed to assist students in A Summer Review of Algebra
COUNCIL ROCK HIGH SCHOOL MATHEMATICS A Note Guideline of Algebraic Concepts Designed to assist students in A Summer Review of Algebra [A teacher prepared compilation of the 7 Algebraic concepts deemed
More information16x y 8x. 16x 81. U n i t 3 P t 1 H o n o r s P a g e 1. Math 3 Unit 3 Day 1 - Factoring Review. I. Greatest Common Factor GCF.
P a g e 1 Math 3 Unit 3 Day 1 - Factoring Review I. Greatest Common Factor GCF Eamples: A. 3 6 B. 4 8 4 C. 16 y 8 II. Difference of Two Squares Draw ( - ) ( + ) Square Root 1 st and Last Term Eamples:
More informationVertex form of a quadratic equation
Verte form of a quadratic equation Nikos Apostolakis Spring 017 Recall 1. Last time we looked at the graphs of quadratic equations in two variables. The upshot was that the graph of the equation: k = a(
More informationChapter 2 Analysis of Graphs of Functions
Chapter Analysis of Graphs of Functions Chapter Analysis of Graphs of Functions Covered in this Chapter:.1 Graphs of Basic Functions and their Domain and Range. Odd, Even Functions, and their Symmetry..
More informationPolynomials and Polynomial Functions
Unit 5: Polynomials and Polynomial Functions Evaluating Polynomial Functions Objectives: SWBAT identify polynomial functions SWBAT evaluate polynomial functions. SWBAT find the end behaviors of polynomial
More informationAlgebra Concepts Equation Solving Flow Chart Page 1 of 6. How Do I Solve This Equation?
Algebra Concepts Equation Solving Flow Chart Page of 6 How Do I Solve This Equation? First, simplify both sides of the equation as much as possible by: combining like terms, removing parentheses using
More information= x. Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions. Math Background
Algebra II Notes Quadratic Functions Unit 3.1 3. Graphing Quadratic Functions Math Background Previousl, ou Identified and graphed linear functions Applied transformations to parent functions Graphed quadratic
More informationMt. Douglas Secondary
Foundations of Math 11 Section 7.1 Quadratic Functions 31 7.1 Quadratic Functions Mt. Douglas Secondar Quadratic functions are found in everda situations, not just in our math classroom. Tossing a ball
More informationLesson 4.1 Exercises, pages
Lesson 4.1 Eercises, pages 57 61 When approimating answers, round to the nearest tenth. A 4. Identify the y-intercept of the graph of each quadratic function. a) y = - 1 + 5-1 b) y = 3-14 + 5 Use mental
More informationChapter Five Notes N P U2C5
Chapter Five Notes N P UC5 Name Period Section 5.: Linear and Quadratic Functions with Modeling In every math class you have had since algebra you have worked with equations. Most of those equations have
More informationSection 2.7 Notes Name: Date: Polynomial and Rational Inequalities
Section.7 Notes Name: Date: Precalculus Polynomial and Rational Inequalities At the beginning of this unit we solved quadratic inequalities by using an analysis of the graph of the parabola combined with
More informationAnswers. Investigation 2. ACE Assignment Choices. Applications. Problem 2.5. Problem 2.1. Problem 2.2. Problem 2.3. Problem 2.4
Answers Investigation ACE Assignment Choices Problem. Core, Problem. Core, Other Applications ; Connections, 3; unassigned choices from previous problems Problem.3 Core Other Connections, ; unassigned
More informationMath-2A Lesson 13-3 (Analyzing Functions, Systems of Equations and Inequalities) Which functions are symmetric about the y-axis?
Math-A Lesson 13-3 (Analyzing Functions, Systems of Equations and Inequalities) Which functions are symmetric about the y-axis? f ( x) x x x x x x 3 3 ( x) x We call functions that are symmetric about
More informationEOC Review. Algebra I
EOC Review Algebra I Order of Operations PEMDAS Parentheses, Eponents, Multiplication/Division, Add/Subtract from left to right. A. Simplif each epression using appropriate Order of Operations.. 5 6 +.
More informationSection 3.3 Graphs of Polynomial Functions
3.3 Graphs of Polynomial Functions 179 Section 3.3 Graphs of Polynomial Functions In the previous section we eplored the short run behavior of quadratics, a special case of polynomials. In this section
More informationPolynomial and Synthetic Division
Polynomial and Synthetic Division Polynomial Division Polynomial Division is very similar to long division. Example: 3x 3 5x 3x 10x 1 3 Polynomial Division 3x 1 x 3x 3 3 x 5x 3x x 6x 4 10x 10x 7 3 x 1
More informationChapter 18 Quadratic Function 2
Chapter 18 Quadratic Function Completed Square Form 1 Consider this special set of numbers - the square numbers or the set of perfect squares. 4 = = 9 = 3 = 16 = 4 = 5 = 5 = Numbers like 5, 11, 15 are
More informationRev Name Date. Solve each of the following equations for y by isolating the square and using the square root property.
Rev 8-8-3 Name Date TI-8 GC 3 Using GC to Graph Parabolae that are Not Functions of Objectives: Recall the square root propert Practice solving a quadratic equation f Graph the two parts of a hizontal
More informationMath 140 Final Sample A Solutions. Tyrone Crisp
Math 4 Final Sample A Solutions Tyrone Crisp (B) Direct substitution gives, so the limit is infinite. When is close to, but greater than,, the numerator is negative while the denominator is positive. So
More informationQUADRATIC FUNCTIONS. ( x 7)(5x 6) = 2. Exercises: 1 3x 5 Sum: 8. We ll expand it by using the distributive property; 9. Let s use the FOIL method;
QUADRATIC FUNCTIONS A. Eercises: 1.. 3. + = + = + + = +. ( 1)(3 5) (3 5) 1(3 5) 6 10 3 5 6 13 5 = = + = +. ( 7)(5 6) (5 6) 7(5 6) 5 6 35 4 5 41 4 3 5 6 10 1 3 5 Sum: 6 + 10+ 3 5 ( + 1)(3 5) = 6 + 13 5
More informationFlash Light Reflectors. Fountains and Projectiles. Algebraically, parabolas are usually defined in two different forms: Standard Form and Vertex Form
Sec 6.1 Conic Sections Parabolas Name: What is a parabola? It is geometrically defined by a set of points or locus of points that are equidistant from a point (the focus) and a line (the directri). To
More informationMath 120. x x 4 x. . In this problem, we are combining fractions. To do this, we must have
Math 10 Final Eam Review 1. 4 5 6 5 4 4 4 7 5 Worked out solutions. In this problem, we are subtracting one polynomial from another. When adding or subtracting polynomials, we combine like terms. Remember
More informationHonors Math 2 Unit 1 Test #2 Review 1
Honors Math Unit 1 Test # Review 1 Test Review & Study Guide Modeling with Quadratics Show ALL work for credit! Use etra paper, if needed. Factor Completely: 1. Factor 8 15. Factor 11 4 3. Factor 1 4.
More informationAdditional Factoring Examples:
Honors Algebra -3 Solving Quadratic Equations by Graphing and Factoring Learning Targets 1. I can solve quadratic equations by graphing. I can solve quadratic equations by factoring 3. I can write a quadratic
More informationExample 1: What do you know about the graph of the function
Section 1.5 Analyzing of Functions In this section, we ll look briefly at four types of functions: polynomial functions, rational functions, eponential functions and logarithmic functions. Eample 1: What
More information3.1 Power Functions & Polynomial Functions
3.1 Power Functions & Polynomial Functions A power function is a function that can be represented in the form f() = p, where the base is a variable and the eponent, p, is a number. The Effect of the Power
More informationFinding the Equation of a Graph. I can give the equation of a curve given just the roots.
National 5 W 7th August Finding the Equation of a Parabola Starter Sketch the graph of y = x - 8x + 15. On your sketch clearly identify the roots, axis of symmetry, turning point and y intercept. Today
More information9.1. 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 informationQUADRATIC EQUATIONS. + 6 = 0 This is a quadratic equation written in standard form. x x = 0 (standard form with c=0). 2 = 9
QUADRATIC EQUATIONS A quadratic equation is always written in the form of: a + b + c = where a The form a + b + c = is called the standard form of a quadratic equation. Eamples: 5 + 6 = This is a quadratic
More information(a) Write down the value of q and of r. (2) Write down the equation of the axis of symmetry. (1) (c) Find the value of p. (3) (Total 6 marks)
1. Let f(x) = p(x q)(x r). Part of the graph of f is shown below. The graph passes through the points ( 2, 0), (0, 4) and (4, 0). (a) Write down the value of q and of r. (b) Write down the equation of
More informationQUADRATIC 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 informationPrecalculus Summer Packet
Precalculus Summer Packet These problems are to be completed to the best of your ability by the first day of school You will be given the opportunity to ask questions about problems you found difficult
More informationChapter 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 informationFinding Slope. Find the slopes of the lines passing through the following points. rise run
Finding Slope Find the slopes of the lines passing through the following points. y y1 Formula for slope: m 1 m rise run Find the slopes of the lines passing through the following points. E #1: (7,0) and
More informationACCUPLACER MATH 0310
The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 00 http://www.academics.utep.edu/tlc MATH 00 Page Linear Equations Linear Equations Eercises 5 Linear Equations Answer to
More informationUnit 5 Solving Quadratic Equations
SM Name: Period: Unit 5 Solving Quadratic Equations 5.1 Solving Quadratic Equations by Factoring Quadratic Equation: Any equation that can be written in the form a b c + + = 0, where a 0. Zero Product
More information2 3 x = 6 4. (x 1) 6
Solutions to Math 201 Final Eam from spring 2007 p. 1 of 16 (some of these problem solutions are out of order, in the interest of saving paper) 1. given equation: 1 2 ( 1) 1 3 = 4 both sides 6: 6 1 1 (
More informationAlgebra Notes Quadratic Functions and Equations Unit 08
Note: This Unit contains concepts that are separated for teacher use, but which must be integrated by the completion of the unit so students can make sense of choosing appropriate methods for solving quadratic
More information+ = + + = x = + = + = 36x
Ch 5 Alg L Homework Worksheets Computation Worksheet #1: You should be able to do these without a calculator! A) Addition (Subtraction = add the opposite of) B) Multiplication (Division = multipl b the
More informationA level Mathematics Student Induction Booklet
Name: A level Mathematics Student Induction Booklet Welcome to A Level mathematics! The objective of this booklet is to help you get started with the A Level Maths course, and to smooth your pathway through
More informationSOLUTIONS FOR PROBLEMS 1-30
. Answer: 5 Evaluate x x + 9 for x SOLUTIONS FOR PROBLEMS - 0 When substituting x in x be sure to do the exponent before the multiplication by to get (). + 9 5 + When multiplying ( ) so that ( 7) ( ).
More informationGraphs of Polynomials: Polynomial functions of degree 2 or higher are smooth and continuous. (No sharp corners or breaks).
Graphs of Polynomials: Polynomial functions of degree or higher are smooth and continuous. (No sharp corners or breaks). These are graphs of polynomials. These are NOT graphs of polynomials There is a
More informationPerforming well in calculus is impossible without a solid algebra foundation. Many calculus
Chapter Algebra Review Performing well in calculus is impossible without a solid algebra foundation. Many calculus problems that you encounter involve a calculus concept but then require many, many steps
More informationALGEBRA II-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION
ALGEBRA II-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION The Quadratic Equation is written as: ; this equation has a degree of. Where a, b and c are integer coefficients (where a 0) The graph of
More informationProperties of Derivatives
6 CHAPTER Properties of Derivatives To investigate derivatives using first principles, we will look at the slope of f ( ) = at the point P (,9 ). Let Q1, Q, Q, Q4, be a sequence of points on the curve
More informationAlgebra I Practice Questions ? 1. Which is equivalent to (A) (B) (C) (D) 2. Which is equivalent to 6 8? (A) 4 3
1. Which is equivalent to 64 100? 10 50 8 10 8 100. Which is equivalent to 6 8? 4 8 1 4. Which is equivalent to 7 6? 4 4 4. Which is equivalent to 4? 8 6 Page 1 of 0 11 Practice Questions 6 1 5. Which
More informationDepartment of Mathematics, University of Wisconsin-Madison Math 114 Worksheet Sections (4.1),
Department of Mathematics, University of Wisconsin-Madison Math 114 Worksheet Sections (4.1), 4.-4.6 1. Find the polynomial function with zeros: -1 (multiplicity ) and 1 (multiplicity ) whose graph passes
More informationSolve Quadratic Equations
Skill: solve quadratic equations by factoring. Solve Quadratic Equations A.SSE.A. Interpret the structure of epressions. Use the structure of an epression to identify ways to rewrite it. For eample, see
More informationLinear Functions A linear function is a common function that represents a straight line
This handout will: Define Linear and Quadratic Functions both graphically and algebraically Examine the associated equations and their components. Look at how each component could affect shape graphically
More informationSecondary 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 informationKCATM 2013 Algebra Team Test. E) No Solution. C x By. E) None of the Above are correct C) 9,19
KCTM 03 lgebra Team Test School ) Solve the inequality: 6 3 4 5 5 3,,3 3, 3 E) No Solution Both and B are correct. ) Solve for : By C C B y C By B C y C By E) None of the bove are correct 3) Which of the
More informationQuadratics NOTES.notebook November 02, 2017
1) Find y where y = 2-1 and a) = 2 b) = -1 c) = 0 2) Epand the brackets and simplify: (m + 4)(2m - 3) To find the equation of quadratic graphs using substitution of a point. 3) Fully factorise 4y 2-5y
More informationUnit 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 informationUnit 2: Functions and Graphs
AMHS Precalculus - Unit 16 Unit : Functions and Graphs Functions A function is a rule that assigns each element in the domain to eactly one element in the range. The domain is the set of all possible inputs
More informationAlgebra Final Exam Review Packet
Algebra 1 00 Final Eam Review Packet UNIT 1 EXPONENTS / RADICALS Eponents Degree of a monomial: Add the degrees of all the in the monomial together. o Eample - Find the degree of 5 7 yz Degree of a polynomial:
More informationSTUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs
STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE Functions & Graphs Contents Functions and Relations... 1 Interval Notation... 3 Graphs: Linear Functions... 5 Lines and Gradients... 7 Graphs: Quadratic
More informationIntroduction. So, why did I even bother to write this?
Introduction This review was originally written for my Calculus I class, but it should be accessible to anyone needing a review in some basic algebra and trig topics. The review contains the occasional
More informationAlgebra Review C H A P T E R. To solve an algebraic equation with one variable, find the value of the unknown variable.
C H A P T E R 6 Algebra Review This chapter reviews key skills and concepts of algebra that you need to know for the SAT. Throughout the chapter are sample questions in the style of SAT questions. Each
More informationChapter XX: 1: Functions. XXXXXXXXXXXXXXX <CT>Chapter 1: Data representation</ct> 1.1 Mappings
978--08-8-8 Cambridge IGCSE and O Level Additional Mathematics Practice Book Ecerpt Chapter XX: : Functions XXXXXXXXXXXXXXX Chapter : Data representation This section will show you how to: understand
More informationx 20 f ( x ) = x Determine the implied domain of the given function. Express your answer in interval notation.
Test 2 Review 1. Given the following relation: 5 2 + = -6 - y Step 1. Rewrite the relation as a function of. Step 2. Using the answer from step 1, evaluate the function at = -1. Step. Using the answer
More informationAlgebra 2 Segment 1 Lesson Summary Notes
Algebra 2 Segment 1 Lesson Summary Notes For each lesson: Read through the LESSON SUMMARY which is located. Read and work through every page in the LESSON. Try each PRACTICE problem and write down the
More informationUsing Intercept Form
8.5 Using Intercept Form Essential Question What are some of the characteristics of the graph of f () = a( p)( q)? Using Zeros to Write Functions Work with a partner. Each graph represents a function of
More informationLesson 7.1 Polynomial Degree and Finite Differences
Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 1 b. 0.2 1. 2 3.2 3 c. 20 16 2 20 2. Determine which of the epressions are polynomials. For each polynomial,
More informationAlgebra I Quadratics Practice Questions
1. Which is equivalent to 64 100? 10 50 8 10 8 100. Which is equivalent to 6 8? 4 8 1 4. Which is equivalent to 7 6? 4 4 4. Which is equivalent to 4? 8 6 From CCSD CSE S Page 1 of 6 1 5. Which is equivalent
More information2-7 Solving Quadratic Inequalities. ax 2 + bx + c > 0 (a 0)
Quadratic Inequalities In One Variable LOOKS LIKE a quadratic equation but Doesn t have an equal sign (=) Has an inequality sign (>,
More informationSection 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 informationAlgebra 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 informationSolving Equations Quick Reference
Solving Equations Quick Reference Integer Rules Addition: If the signs are the same, add the numbers and keep the sign. If the signs are different, subtract the numbers and keep the sign of the number
More informationMath 150: Intermediate Algebra Spring 2012 Fall 2005 : Mock Exam 4 (z z
Math 150: Intermediate Algebra Spring 01 Fall 005 : Mock Eam 4 (z 9. - z NAME 10.6) TICKET # [ 13 problems: 10 points each ] Answer problems and epress your answers appropriately, that is ; simply state
More information3 Polynomial and Rational Functions
3 Polnomial and Rational Functions 3.1 Quadratic Functions and Models 3.2 Polnomial Functions and Their Graphs 3.3 Dividing Polnomials 3.4 Real Zeros of Polnomials 3.5 Comple Zeros and the Fundamental
More informationGoal: To graph points in the Cartesian plane, identify functions by graphs and equations, use function notation
Section -1 Functions Goal: To graph points in the Cartesian plane, identify functions by graphs and equations, use function notation Definition: A rule that produces eactly one output for one input is
More informationLesson 9 Exploring Graphs of Quadratic Functions
Exploring Graphs of Quadratic Functions Graph the following system of linear inequalities: { y > 1 2 x 5 3x + 2y 14 a What are three points that are solutions to the system of inequalities? b Is the point
More informationC. Finding roots of trinomials: 1st Example: x 2 5x = 14 x 2 5x 14 = 0 (x 7)(x + 2) = 0 Answer: x = 7 or x = -2
AP Calculus Students: Welcome to AP Calculus. Class begins in approimately - months. In this packet, you will find numerous topics that were covered in your Algebra and Pre-Calculus courses. These are
More information(b) Equation for a parabola: c) Direction of Opening (1) If a is positive, it opens (2) If a is negative, it opens
Section.1 Graphing Quadratics Objectives: 1. Graph Quadratic Functions. Find the ais of symmetry and coordinates of the verte of a parabola.. Model data using a quadratic function. y = 5 I. Think and Discuss
More informationSCIE 4101 Fall Math Review Packet #2 Notes Patterns and Algebra I Topics
SCIE 4101 Fall 014 Math Review Packet # Notes Patterns and Algebra I Topics I consider Algebra and algebraic thought to be the heart of mathematics everything else before that is arithmetic. The first
More informationWorksheets for GCSE Mathematics. Quadratics. mr-mathematics.com Maths Resources for Teachers. Algebra
Worksheets for GCSE Mathematics Quadratics mr-mathematics.com Maths Resources for Teachers Algebra Quadratics Worksheets Contents Differentiated Independent Learning Worksheets Solving x + bx + c by factorisation
More informationMath 2412 Activity 2(Due by EOC Feb. 27) Find the quadratic function that satisfies the given conditions. Show your work!
Math 4 Activity (Due by EOC Feb 7) Find the quadratic function that satisfies the given conditions Show your work! The graph has a verte at 5, and it passes through the point, 0 7 The graph passes through
More information