2 P a g e. Essential Questions:

Transcription

1 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 Polynomials Graphing Quadratic Functions & Characteristics of Quadratic Functions Application of Quadratic Functions Quadratic Regression and Choosing a Model of Best Fit Review & Practice Test Common Core Standards NC.M1.A- APR.1 NC.M1.A- APR.3 NC.M1.A- CED.2 NC.M1.A- REI.1 NC.M1.A- REI.11 NC.M1.A- REI.4 NC.M1.A- SSE.1a NC.M1.A- SSE.1b NC.M1.A- SSE.3 NC.M1.F- BF.1b NC.M1.F- IF.2 NC.M1.F- IF.4 Build an understanding that operations with polynomials are comparable to operations with integers by adding and subtracting quadratic expressions and by adding, subtracting, and multiplying linear expressions. Understand the relationships among the factors of a quadratic expression, the solutions of a quadratic equation, and the zeros of a quadratic function. Create and graph equations in two variables to represent linear, exponential, and quadratic relationships between quantities. Justify a chosen solution method and each step of the solving process for linear and quadratic equations using mathematical reasoning. Build an understanding of why the x-coordinates of the points where the graphs of two linear, exponential, and/or quadratic equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x) and approximate solutions using graphing technology or successive approximations with a table of values. Solve for the real solutions of quadratic equations in one variable by taking square roots and factoring. Interpret expressions that represent a quantity in terms of its context. a) Identify and interpret parts of a linear, exponential, or quadratic expression, including terms, factors, coefficients, and exponents. Interpret expressions that represent a quantity in terms of its context. b) Interpret a linear, exponential, or quadratic expression made of multiple parts as a combination of entities to give meaning to an expression. Write an equivalent form of a quadratic expression, ax2 + bx+ c, where a is an integer, by factoring to reveal the solutions of the equation or the zeros of the function the expression defines. Write a function that describes a relationship between two quantities. b) Build a function that models a relationship between two quantities by combining linear, exponential, or quadratic functions with addition and subtraction or two linear functions with multiplication. Use function notation to evaluate linear, quadratic, and exponential functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities, including: intercepts; intervals where the function is increasing, decreasing, positive, or negative; and maximums and minimums. 1 P a g e

2 I understand the effect of gravity controls the a value of a quadratic in standard form when modeling projectile motion. initial height controls the c value of a quadratic in standard form when modeling projectile motion. initial velocity controls the b value of a quadratic in standard form when modeling projectile motion. that the x-intercepts of quadratic functions are the solutions of quadratic equations. that the rate of change of a quadratic function does not remain constant. that the x-intercepts/solutions/zeros/roots can be determined by factoring for some quadratic functions. I can use function notation to evaluate a quadratic function given a value in the domain. interpret the contextual meaning of a given point from a quadratic function in function notation. interpret the meaning of the independent and dependent variables in context of a quadratic function. interpret and analyze key features of a quadratic function in context including positive/negative, increasing/decreasing, intercepts, maximum/minimum and domain/range when given the function as a table, graph, and/or verbal description. identify the terms, factors and coefficients of a quadratic expression. interpret the terms, factors and coefficients of a quadratic expression in terms of the context. create an equation in two variables to represent a quadratic relationship between two quantities. graph a quadratic equation that represents a relationship between two quantities. choose an appropriate domain and range for a quadratic function. identify the maximum and minimum of quadratic functions. identify where a quadratic function is increasing and decreasing. compare linear and quadratic functions symbolically, graphically, verbally, and using tables. build a quadratic function by multiplying linear equations and combining two quadratic equations with addition and subtraction. Essential Questions: How can projectile motion be modeled using a quadratic function? How does knowing the definition of a maximum or minimum help you visualize the graph of a quadratic function? How do you determine which solution to use for a quadratic equation? How is factoring connected to the distributive property? How can I compare operations with integers to operations with quadratic expressions? What types of information are contained in various forms of a quadratic function? Vocabulary: axis of symmetry binomial constant degree of a monomial degree of a polynomial difference of squares extreme values factoring intercepts linear expression monomial polynomial relative minimum or maximum solutions or roots of a quadratic equation standard form of a polynomial symmetry trinomial vertex x-intercepts of a quadratic function 2 P a g e

3 Algebra Tiles x 2 = x 2 = x = x = +1 = 1 = You simplify variable expressions by combining like terms. Use zero pairs, where needed, and perform as many operations as possible within the expression. Example. 3a + ( a) = + = = 2a 0 2y y 3 = + + = 3y Use models to simplify these expressions. 1. 4x + 7x = 2. 5b 2b = 3. 3a + (-2a) = 4. 5y ( 2y) = 5. 3c c 2 = x + 3x x = Remember 2(x + 3) means + = 2x + 6 Use models to simplify these expressions. 7. 2(y 2 4) = 8. 3(a + 3) = 9. 2(z 4) + 9 = 10. 2(2b 2 1) + 7 = c 2 + 4(c 2 + 3c) 5c = 12. 8y + 2(3 y) 4 = 3 P a g e

4 What are polynomials? A polynomial is a monomial or the sum of monomials. A monomial is a number, a variable, or a product of numbers and variables. After being simplified, a polynomial can be named based on its degree and its number of terms. The degree of a monomial is the sum of the exponents of the monomial s variables. The degree of a polynomial is equal to the degree of the monomial term with the largest degree. Polynomial Degree Number of Name Terms 3x linear binomial Graph a 2 3a quadratic trinomial 6z cubic monomial 2x 4 + 2x th degree trinomial constant monomial (or just constant ) Polynomials are usually written in standard form. The standard form of a polynomial is when the polynomial is written such that the degrees of the monomial terms decrease from left to right. 4 P a g e

5 What is the degree of each monomial? 1. 3ab x a 2 b Write each polynomial in standard form and then name it. 5. 3x 2x 2 + 3x x x 8. 5b 2 + b 3 2b x 3x 2 + 2x x 2 Modeling Polynomials with Algebra tiles Add and subtract by combining like terms and by using adding zero where needed. Ex. (2x 2 + x + 3) + (x 2 3x + 1) + Form a zero pair with +x and x. = 3x 2 2x + 4 Try these: 0 1. (2x 2 + 3x + 3) + (x 2 + 2x 2) 2. (3x 2 + 4x 3) + (4x 2 2x 4) 3. (x 2 + 3x 4) (2x 2 + 2x 3) 4. (5x 2 2x + 1) (x 2 +2x 4) 5 P a g e

6 EXPLORING MULTIPLYING POLYNOMIALS x 2 = x 2 = x = x = +1 = 1 = ex. 1. x(x + 1) 2. 2x( x + 3) x + 1 x 2 x x x x x 2 x = x 2 + x = 2x 2 + 6x x 2 x x x Try these examples: 1. 2(3x + 1) 2. x(x + 2) 3. 2x(x + 2) 6 P a g e

7 Homework Simplify each sum or difference. Write your answer in standard form. Name each polynomial. (To avoid sign errors, carefully use the definition of subtraction to rewrite all subtraction as addition, and carefully use the distributive property when a polynomial with more than one term is being subtracted.) 1. (3x 2 2x 7) + (5x 2 7x + 3) 2. (8x 2 10) (3x 2 5x 9) 3. (6t 5 4t 2 + 7) (9t 4 2t 5 2t) 4. (11 2a + 3a 3 ) + (7 a 4 5a + 7a 2 ) 5. (3n 3 2n 4 n) (5n 5 n 2 + 7) 6. x(2x 1) 7. 3x(2x + 1) 8. 2x( x + 4) 9. -x ( -3x + 2) 10. challenge 4x 2 (x + 4) 7 P a g e

8 ex. 1. (x + 1)(x + 1) 2. (x + 1)( x + 3) x + 1 -x + 3 x x 2 x x x x x 2 x = x 2 + 2x + 1 = x 2 + 2x x x 1 Draw out the algebra tiles 1. (x + 2)(x + 1) 2. (x + 2)(x + 2) 3. ( 2x 1)(x + 2) 8 P a g e

9 Multiplying Binomials: The Distributive Property *Without really thinking about it, how would you simplify (a + b) 2? To calculate the product of (2x + 4) and (3x + 5), you can use a rectangle to represent the problem. You can then calculate the area of each part and then calculate the sum of all of the parts to find 2x + 4 the area of the whole rectangle. 3x +5 2x + 4 3x 6x 2 12x +5 10x 20 (2x + 4)(3x + 5) = 6x x + 10x + 20 = 6x x + 20 You can also use the distributive property without the graphic organizer. (2x + 4)(3x + 5) = (2x)(3x) + (2x)(5) + (4)(3x) + 4(5) = 6x x + 20 A mnemonic often used when multiplying two binomials is FOIL. FOIL is an acronym for First, Outer (or outside), Inner (or inside), and Last. In the example (2x + 4)(3x + 5) F: (2x)(3x) = 6x 2 ; O: (2x)(5) = 10x ; I: (4)(3x)=12x ; L: (4)(5) = 20 If the two polynomial factors are not both binomials, you cannot use the FOIL method. When you use the distributive property, a box, or area, diagram can help you organize your work. When multiplying polynomials, think about the distributive property very carefully. Pay attention to the sign of each term. 1. (2a 1)(3a + 7) 2. (3a 7)(a 2) 3. (x + 1)(x + 4) 4. (7n 5)(5n + 4) F: F: F: F: O: O: O: O: I: I: I: I: L: L: L: L: 9 P a g e

10 5. (x 3)(x 2 2x + 5) 6. (y + 6)(y 2 + 4y 3) 7. (5x 2 3)(5x 2 + 3) 8. (x 2 2)(x + 3) 9. (2a 2 + 4)(a + 7) 10. (2b 2 7)(b + 3b 2 ) 11. (9w 2 + 5)(2w 3 + 3w + 1) 12. (y 2)(3y 2 + 6y 7) 13. Mr. Wilson has two congruent, rectangular gardens 3 feet apart from each other. The length of each garden is 4 times its width. There is a 3-ft wide walkway around each garden. 3ft 3ft a. Write an expression for the combined areas of the gardens and walkway. 3ft 3ft 3ft 3ft 3ft b. Write an expression for the area of the walkway only. c. The walkway s area is 150 square feet. How big is each garden? 10 P a g e

11 14. Calculate the surface area of the space figure. (x + 1) cm (3x 2) cm (4x 3) cm (2x + 4) cm 15. Calculate the volume of the space figure. Complete each area diagram. Write down the problem that it illustrates and simplify the product x 5 3x 7 5 8x 2 6x 2 5x 15x 2 3 6x 12x 3 These 3 products deal with special binomial patterns a b a b a b a a a b b b 11 P a g e

12 Binomials: Multiplying Special Cases Use the algebra tiles to model (x + 3) 2 Use the algebra tiles to model x Draw below Draw below Write the algebraic equation for (x+3) 2 Write the algebraic equation for x Are the results the same Why??? A) The Square of a Binomial: (a + b) 2 = a 2 + 2ab + b 2 (a b) 2 = a 2 2ab + b 2 The square of a binomial is the square of the first term plus twice the product of the two terms plus the square of the last term. Its product is called a perfect square trinomial. B) The Difference of Squares (a + b)(a b) = a 2 b 2 The product of the sum and difference of the same two terms is the difference of their squares. (a + b) and (a b) are conjugates of one another. You can always use the distributive property to multiply these two special cases, but you will be using them a lot, so you should have them memorized and mastered. 12 P a g e

13 Simplify each product. 1. (x 2)(x + 2) 6. (x + 7) 2 2. (x 2) 2 7. (3x 9)(3x + 9) 3. (x + 2) 2 8. (3x 9) 2 4. (x 7)(x + 7) 9. (3x + 9) 2 5. (x 7)(x 7) 10. Identify each polynomial as a difference of squares, a perfect square trinomial, or neither. a. 8x 2 9 b. 25x 2 20x 4 c. x 2 1 d. 4x e. x 2 + 6x P a g e

14 Practice Multiplying Polynomials Simplify each product. Circle the ones that are one of the two patterns. Time yourself. 14 P a g e

15 Factoring Quadratic Polynomials In order to factor polynomials, put your algebra tiles into a rectangular array. You may need to add zero pairs to complete your model. Ex. Factor x 2 + 4x + 3 Rearrange What should be on the left hand side? above? x 2 + 4x + 3 = (x + 1)(x + 3) Factor 2x 2 + 3x Rearrange 2x 2 + 3x = x(2x + 3) Factor x 2 + 3x + 4 Rearrange Notice I had to add a zero -pair (x & x). x 2 + 3x + 4 = (x + 1)( x + 4) 15 P a g e

16 Factor each problem. Draw the algebra tiles 1. 3x 2 + 2x 5. x 2 6x x 2 15x y 2 2y x 2 + 6x m 2 + m a 2 + 4a x 2 + 7x P a g e

17 Factoring without algebra tiles In the first example, we ll just look at the first and last steps of factoring polynomials. Step 1: Factoring out the GCF The greatest common factor, or GCF, is the largest factor the terms have in common. Factoring a polynomial reverses the multiplication process. To find the GCF, list all the prime factors of each term and then circle or underline the factors common to ALL terms. If the coefficient of the first term of a polynomial written in standard form is negative, factor out 1. Example 1: What is the GCF of 12x x 2 15x? Calculate the GCF of the 3 terms 12x 3 : 2 * 2 * 3 * x * x * x 18x 2 : 2 * 3 * 3 * x * x 15x: 3 * 5 * x So, the GCF is 3x Use the distributive property to write the product. 3x (4x 2 + 6x 5) To finish the problem, check each factor to make sure that it cannot be factored any further. Multiply the factors to make sure it was factored correctly. Sign errors and arithmetic errors are a fairly common errors, so pay close attention to your signs and double check your multiplication to make sure you didn t add. Note: When finding the GCF of the variables, find the least power of each variable. For example: x 5 y x 3 y 2 + x 2 y 4 x 2 is the least power of x y is the least power of y GCF is x 2 y Final factored answer is x 2 y (x 3 xy + y 3 ) Factor the following polynomials by factoring out the GCF. 1) 5x x 3 2) 4b b 2 8b 3) 3d 3 12d d 17 P a g e

18 Step 2: Check for the two patterns If the polynomial is a binomial, the only way you have learned to factor it so far is to factor out the GCF and then check to see if it is a difference of squares. In a later math course, you will be taught how to factor a binomial if it is a sum or a difference of cubes. If the polynomial is a trinomial, after factoring out the GCF, check to see if it is a perfect square trinomial. If it is, you can factor it quickly as the square of a binomial. a 2 + 2ab + b 2 = (a + b) 2 a 2 2ab + b 2 = (a b) 2 a 2 b 2 = (a + b)(a b) 4) x ) 4x ) x 2 6x + 9 7) 12x x x Step 3: Factoring Trinomials ax 2 + bx + c Factoring polynomials can help you solve many different types of problems. In Math I, you will use the factors of quadratic polynomials to help you graph quadratic functions and solve problems modeled by quadratic functions. Not all quadratic polynomials are factorable, but most of the ones you deal with this year will be. After factoring out the GCF, if you are left with a quadratic trinomial that is a not perfect square trinomial, you may have to do a little bit of guessing and checking to calculate its factors. You ve already learned that the product of two linear binomials is a quadratic trinomial. Use that knowledge to learn how to factor quadratic trinomials. Using variables to represent coefficients and constants. Starting with the quadratic trinomial Ax 2 + Bx + C, we ll assume that the GCF has already been factored out, so A is a counting number, and B & C are integers. We can declare its linear binomial factors as (Dx + E) and (Gx + H). D and G must be counting numbers, since A is a counting number. Remember to factor out a 1 if the quadratic term has a negative coefficient. E and H are the constant terms will be integers. Gx Dx Ax 2 = GDx 2 + E GEx We can use an area model to look for patterns that will help us factor quadratic trinomials. +H DHx EH = C The factors of A will always be G and D. The factors of C will always be E and H. DHx + GEx = (DH + GE)x = Bx Because the GCF was already factored out of Ax 2 + Bx + C, G and H share no common factors other than 1, and D and E share no common factors other than P a g e

19 A key characteristic that will help you factor quadratic trinomials more quickly is that the factors of AC have a sum of B. AC = (GD)(EH). The factors could also be written as (DH)(GE). If you add DH and GE, you get the value of B. Another key characteristic deals with the signs of the constants. If C is positive, E and H must be the same sign. If C is negative, E and H must have different signs. Before you start factoring a quadratic trinomial, remember to do the following: 1. Factor out the GCF. If the coefficient of the quadratic term is negative, factor 1 out of the quadratic trinomial. 2. Check to see if the trinomial is a perfect square trinomial. If it is, write its factors as the square of a binomial. Sometimes, the coefficient of the quadratic term will be 1. These quadratic trinomials can be factored fairly quickly. Since A = 1, we have a quadratic trinomial in the form of x 2 + Bx + C. From the area model, we know that the factors of A and C must have a sum of B. When A = 1, the factors of C must equal B. G and D must both be equal to 1 since A is 1 and we only use integers when we factor. Example: Factor completely. x 2 7x -30 Draw a 2x2 box: ac b Place the highest degree term in the top left box and the constant in the bottom right box. Use the diamond to find the other two boxes. Find 2 terms that multiply to get ac and add/subtract to get b. Find the GCF of the 1 st row to find the 1 st part of the factor on the left. Find the top factor Find the 2 nd factor on the left 19 P a g e

20 8) x 2 + 7x ) g 2 + 7g ) d 2 17d + 42 Some polynomials have two variables. If they are factorable, each binomial factor will have two linear terms instead of one linear term and one constant term. 11) x xy + 24y 2 12) 2m 2 34mn 120n 2 If the coefficient of the quadratic term isn t 1, you can use the pattern we looked at earlier to help you factor it. If graphic organizers help you, draw the area model. 13) 10x x ) 10x x 6 Step 4: Check each factor again to make sure the polynomial is fully factored. Multiply your factors to make sure you factored correctly. Even though we haven t yet covered steps 2 and 3, we ll look at the final step. Every time you fully factor a polynomial, you should check each factor when you re done to make sure you didn t miss any factors. You should also multiply your factors to make sure the product is correct. 20 P a g e

21 Fully factor each quadratic polynomial. Practice following the 4 steps. 1. x 2 3x x x x 2 10x x 2 8x w 2 + w 6 6. y 2 21y x 2 4x 5 8. x x x 2 15x x 2 11x w w y 2 3y m 2 13mn + 42n x 2 + 2xy 15y a 2 3ab 10b x xy + 72y x xy + 72y x 2 xy 72y x 2 21xy 72y x 2 12x x 2 12x x 2 21x w 2 29w y 2 2y x 2 6x x x 2 12x P a g e

22 Fully factor each polynomial. Practice following the 4 steps. 1. 6a a a 2 13a x 2 + 5x n 2 + 3n n 2 7n n n n 2 11n n 2 8n x 2 4x y 2 25y x x x x y 2 12y x 2 + 9x x x y 2 16y x 2 + 4x x 2 4x y 2 y x x x x x x 2 + 6x x 2 3x 25. 4x 2 + 8x x x x 2 20x x x 2 56x P a g e

23 More Practice Factoring Polynomials Fully factor each polynomial. Circle the ones that can be factored by using one of the two patterns. Time yourself. 23 P a g e

24 Properties of Parabola A is a function that can be written in the Standard Form of y ax bx c where a, b, and c are real numbers and a 0. Ex: y 5x y 2x 7 2 y x x 3 The domain of a quadratic function is. The graph of a quadratic function is a U-shaped curve called a. All parabolas have a, the lowest or highest point on the graph (depending upon whether it opens up or down). The is an imaginary line which goes through the vertex and about which the parabola is symmetric. Characteristics of the Graph of a Quadratic Function: 2 y ax bx c Direction of Opening: When a 0, the parabola opens : 24 P a g e When a 0, the parabola opens : Stretch: When a 1, the parabola is vertically. When a 1, the parabola is vertically. Axis of symmetry: This is a vertical line passing through the vertex. Its equation is. Vertex: The highest or lowest point of the parabola is called the vertex, which is on the axis of symmetry. To find the vertex, plug in b x 2a and solve for y. This yields a point (, ) x-intercepts: are the 0, 1, or 2 points where the parabola crosses the x-axis. Plug in y = 0 and solve for x. y-intercept: is the point where the parabola crosses the y-axis. Plug in x = 0 and solve for y: y = c.

25 Without graphing the quadratic functions, complete the requested information: 1.) f x x x 2 ( ) What is the direction of opening? Is the vertex a max or min? Compare to y = x 2? 5 2 g( x) x x 3 2.) 4 What is the direction of opening? Is the vertex a max or min? Compared to y = x 2? 3.) The parabola y = x 2 is graphed to the right. Note its vertex (, ) and its width. You will be asked to compare other parabolas to this graph. Graphing in STANDARD FORM ( 2 y ax bx c ): we need to find the vertex first. Vertex - list a =, b =, c = b - find x = 2 a - plug this x-value into the function (table) - this point (, ) is the vertex of the parabola Graphing - put the vertex you found in the center of your x-y chart. - choose 2 x-values less than and 2 x-values more than your vertex. - plug in these x values to get 4 more points. - graph all 5 points Find the vertex of each parabola. Graph the function and find the requested information 4.) f(x)= -x 2 + 2x Vertex: Max or min? Direction of opening? Axis of symmetry: y x 5.) h(x) = 2x 2 + 4x Vertex: Max or min? Direction of opening? Axis of symmetry: y x 25 P a g e

26 Compare to the graph of y = x 2? Compare to the graph of y = x 2? Unit 6 Skills (Review & Preview) naming polynomials simplifying polynomials: standard form, adding, subtracting and multiplying including the 2 special cases factoring polynomials including the 2 special cases calculating zeros of quadratic functions calculating the axis of symmetry using the zeroes(roots or x-intercepts) of a quadratic function calculating the vertex using the axis of symmetry of a quadratic function identifying quadratic functions by looking at the characteristics of a graph without graphing, using the coefficient of the quadratic term to determine whether the parabola has a maximum vertex and opens down or a minimum vertex and opens up as well as determining how wide or narrow it is compared to other quadratic functions graphing a quadratic function by using the characteristics as well as graphing it by using substitution calculating the y-intercepts of a quadratic function 1. What is the standard form of a quadratic function? 2. What is the parabola s vertex and where is it located? State whether the parabola has a minimum or maximum point and draw its axis of symmetry (dashed line). 26 P a g e

27 What is the vertex? Compare the value of a to zero. What is the vertex? Compare the value of a to zero. What is the vertex? Compare the value of a to zero. 12. Order the quadratic functions from widest to narrowest graphs. f(x) = 3x 2 2x + 3 ; g(x) = 5x 2 ; h(x) = 6x 2 + 2x 1 ; j(x) = 1 2 x2 ; k(x) = x 2 13: Use the characteristics of a quadratic function to calculate the vertex & 2 points on each side of the vertex. Sketch the parabola. y = x a. Factor the polynomial. Solve for the zeroes/roots of the function. y b. What is the axis of symmetry? Graph it. c. What is the vertex? Graph it x d. What is the y-intercept? Graph it and its reflection over the axis of symmetry. e. Calculate one other point on the graph. Graph it and its reflection over the axis of symmetry. 14. y = x 2 Vertex: minimum or maximum 15. y = 2x 2 5x 1 Vertex: minimum or maximum 27 P a g e

28 Width compared to y = x 2 : wider, narrower or equal Parabola opens: up or down y-intercept: Width compared to y = x 2 : wider, narrower or equal Parabola opens: up or down y-intercept: 1. y = x 2 4x + 3 a) What is the axis of symmetry? Graph it. b) What is the vertex? Graph it c) What is the y-intercept? Graph it and its reflection over the axis of symmetry. d) Find one other point on the graph. Graph it and its reflection over the axis of symmetry. e) Sketch the parabola. 28 P a g e

29 2. y = x 2 + 2x 1 a) What is the axis of symmetry? Graph it. b) What is the vertex? Graph it c) What is the y-intercept? Graph it and its reflection over the axis of symmetry. d) Find one other point on the graph. Graph it and its reflection over the axis of symmetry. e) Sketch the parabola. 3. y = x 2 + 2x a) What is the axis of symmetry? Graph it. b) What is the vertex? Graph it c) What is the y-intercept? Graph it and its reflection over the axis of symmetry. d) Find one other point on the graph. Graph it and its reflection over the axis of symmetry. e) Sketch the parabola. 4. y = x 2 2x + 3 a) What is the axis of symmetry? Graph it. b) What is the vertex? Graph it c) What is the y-intercept? Graph it and its reflection over the axis of symmetry. d) Find one other point on the graph. Graph it and its reflection over the axis of symmetry. e) Sketch the parabola. 29 P a g e

30 Applications of Quadratics Examples Only ask 4 questions: a) zeros b) max/min height (y value of the vertex) c) when it hits the max/min (x value of the vertex) d) height when x = 0 (y intercept) 1. Marta throws a baseball with an initial upward velocity of 70 feet per second. This equation h(t) = 16t t models the situation. a. Ignoring Marta s height, how long after she releases the ball, will it hit the ground? b. What is the maximum height of the baseball? c. When does the ball reach the maximum height? 2. A rectangular lot is 50 feet wide and 60 feet long. If both the width and the length are increased by the same amount, the area is increased by 1200 square feet. Find the amount by which both the width and the length are increased. 3. A rectangular lawn has dimensions of 24 feet by 32 feet. A sidewalk will be constructed along the inside edges of all four sides. The remaining lawn will have an area of 425 square feet. How wide is the walk? 4. The floor of a rectangular cage has a length 4 feet greater than its width, w. Bob will make a new cage and increase each dimension of the cage s floor by 2 feet. a. Write an expression to represent the area of the new cage s floor in terms of the width of the original cage s floor. b. Compare the volume of the new cage to the volume of the original cage. 30 P a g e

31 Applications HW 1. A volcanic eruption blasts a boulder upward with an initial velocity of 240 feet per second. This is modeled by the equation h(t) = 16t t. a. How long will it take the boulder to hit the ground? b. How high was the boulder after 5 seconds? 2. A rectangular lawn is 60 feet by 80 feet. How wide of a uniform strip must be cut around the edge when mowing the grass in order for half of the lawn to be cut? 3. A picture frame measures 12 cm by 20 cm (uniform width). The picture without the frame measures 84 Find the width of the frame. 2 cm. 4. A rectangular picture is 12 inches by 16 inches. If a frame of uniform width contains an area of 165 square inches (frame only), what is the width of the frame? 5. A farmer has 400 feet of fencing. He wants to fence off a rectangular field that borders a straight river (no fence along the river). What are the dimensions that will give him the largest area? 6. A rectangular pen is to be built along a wall from 72 yards of fencing. Find the maximum area that can be enclosed and the dimensions of the pen. 7. A rectangular piece of ground is to be enclosed on three sides by 160 ft of fencing. The fourth side is the barn. Find the dimensions of the enclosure and the maximum area that can be enclosed. 31 P a g e

32 8. The length of a tropical garden at a local conservatory is 5 feet more that its width. A walkway 2 feet wide surrounds the outside of the garden. If the total area of the walkway and garden is 594 square feet, find the dimensions of the garden. 9. Helen has a rectangular garden 25 feet by 50 feet. She wants to increase the garden on all sides by an equal amount. If the area of the garden will be increased by 400 square feet, by how much will each dimension be increased? 10. A rectangle is 5 centimeters longer than it is wide. Find the possible dimensions if the area of the rectangle is 104 square centimeters. 11. The YMCA has a 40 foot by 60 foot area in which to build a swimming pool. The pool will be surrounded by a concrete sidewalk of uniform width. What could the width of the sidewalk be if organizers want the pool to be 1500 square feet? 12. The function f(t) = 5t t + 60 models the approximate height of an object in feet t seconds after it is launched. (Challenge) a. How many seconds does it take the object to hit the ground? b. How high off the ground will the object get? c. How long after the launch will the object be at its maximum height? 32 P a g e

33 Using Regression to Calculate a Function Rule for Data (Linear, Exponential, and Quadratic) For each of the following, type in the data into a graphing calculator. Study the data and decide which regression is the most appropriate. Use the appropriate regression program on the calculator to find the function rule for the data. 1. x y Type: Function rule: 2. x y Type: Function rule: 3. x y Type: Function rule: 4. x y Type: Function rule: 5. x y Type: Function rule: 6. x y Type: Function rule: When the x-values increase by 1, there are some quick calculations that can be done to see whether the data is linear, exponential, or quadratic. As x-values increase by 1, if the y-values have a common difference, the data is linear. As the x-values increase by 1, if the y-values have a common ratio, the data is exponential. As the x-values increase by 1, if the y-values have a common second difference (the difference of the differences), the data is quadratic. 33 P a g e

34 Unit 6 Practice Assessment 1. Name each polynomial based on its degree and number of terms. 7x 3 1 2x 2 x 2 9 7x 8 + 5x 3 4x 3 2. Write each polynomial in standard form. 5x x 2 2x + 4x = 3 11x 9x = 2 3x 3 2x x + 8x = 6x 2 6x + 2x x 2 2x + x 3 = Calculate the sum or difference. Write the answer in standard form. 3. (3x 2 3x + 5) (6x 2 + 6x 9) = 4. (2x 4 + 5x 3 + 5) + (3x 4 7x 3 5x 2 6x) = Simplify each product. Write your answer in standard form. Show your work. 5. 4x(3 7x) = 6. (2x + 7)(2x 7) = 7. (5x + 2)(5x + 2) = 8. (4x + 3)(2x 5) = Fully factor each polynomial. 9. 4x 2 9 = x x + 81 = 11. x 2 2x 15 = 12. 4x 2 + 4x 3 = 13. Calculate the zeroes of the quadratic functions. Use your work from #11 & #12. Graph the zeroes and list their coordinates. y = x 2 2x 15 y = 4x 2 + 4x 3 34 P a g e

35 14. A quadratic function has zeroes at (2, 0) and (4, 0). What is the function s axis of symmetry? A quadratic function has zeroes at (7, 0) and ( 3, 0). What is the function s axis of symmetry? Sketch the function. Your sketch must include the vertex and two points on each side of it. 15. y = x 2 + 4x 16. y = x 2 4x What is the axis of symmetry for #15? What is the axis of symmetry for # 16? 18. What is the vertex for #15? What is the vertex for # 16? Use the function rules for #19 & #20. f(x) = x 2 3x 4 ; g(x) = 6x 2 ; h(x) = x 2 + 2x + 5 ; j(x) = 5x 2 2x Which of the functions {f(x), g(x), h(x), and j(x)} open up? Which of the functions {f(x), g(x), h(x), and j(x)} open down? Which of the functions {f(x), g(x), h(x), and j(x)} have a maximum vertex? Which of the functions {f(x), g(x), h(x), and j(x)} have a minimum vertex? Which of the functions {f(x), g(x), h(x), and j(x)} is the narrowest? 20. What are the y-intercepts for each of the functions? f(x): g(x): h(x): j(x): 35 P a g e