Chapter 6: Quadratic Functions & Their Algebra Topics: 1. Quadratic Function Review. Factoring: With Greatest Common Factor & Difference of Two Squares 3. Factoring: Trinomials 4. Complete Factoring 5. Factoring by Grouping 6. Factoring: Special Cases
LESSON 1 QUADRATIC FUNCTION REVIEW Linear and exponential functions are used throughout mathematics and science due to their simplicity and applicability. Quadratic functions comprise another very important category of functions. You studied these extensively in Common Core Algebra I, but we will review many of their important characteristics in this unit. QUADRATIC FUNCTIONS Any function of the form f x ax bx c where the leading coefficient, a, is not zero. Exercise #1: Without the use of your calculator, evaluate each of the following quadratic functions for the specified input values. Recall that, according to the formal Order of Operations, exponent evaluation should always come first. (a) f x x (b) g x x 5 (c) h x x 4x f 3 g h f 5 g 1 h 3 Graphs of quadratic functions form what are known as parabolas. The simplest quadratic function, and one that you should be very familiar with, is reviewed in the next exercise. Exercise #: Consider the simplest of all quadratic functions y x. (a) Create a table of values to plot this function over the domain interval 3 x 3. y x 3 1 0 1 3 y x (b) Sketch a graph of this function on the grid to the right. (c) State the coordinates of the vertex of this parabola. x (d) State the equation of this parabola s axis of symmetry. (e) Over what interval is this function increasing?
All quadratic functions that have unlimited domains (domains that consist of the set of all real numbers) have a vertex (turning point) and an axis of symmetry. It is important to be able to sketch a parabola using your graphing calculator to generate a table of values. f x x 6x 5. Exercise #3: Consider the quadratic function (a) Using a TABLE on your graphing calculator, determine the vertex of this function. (b) What is the range of this quadratic? y (c) Graph this function on the grid to the right. Use your vertex as a middle value of your graph. Graph a few in each direction from there. (d) Why does this parabola open downward as opposed to y x which opened upward? (e) Between what two consecutive integers does the larger solution to the equation x 6x 5 0 lie? Show this point on your graph. x Exercise #4: A sketch of the quadratic function y x 11x 6 is shown below marked with points at its intercepts and its vertex. Using tables or a graph on your calculator, determine the coordinates for each of the points. y The x-intercepts: A B (Zeroes) The y-intercept: D The vertex: C A B x Over what interval is this function positive? D C
LESSON 1 HOMEWORK: QUADRATIC FUNCTION REVIEW FLUENCY 1. Without the use of your calculator, evaluate each of the following quadratic functions for the specified input values. (a) g x x 9 (b) f x x 8x (c) h x x x 6 g 5 f 3 h 0 g 3 f 1 h. Which of the following represents the y-intercept of the quadratic function f(x) = x 7x + 9? (Recall, that the y-intercept of a graph always occurs at f(0).) (1) 7 (3) 7 () (4) 9 3. For a particular quadratic function, the leading coefficient is negative and the function has a vertex whose coordinates are 3,14. Which of the following must be the range of this quadratic? (1) y y 3 (3) y y 14 () y y 3 (4) y y 14 4. A parabola has one x-intercept of x and an axis of symmetry of x 4. Which of the following represents its other x-intercept? (Hint, think of how far the given x-intercept is away from the axis.) (1) x 3 (3) x 6 () x 10 (4) x 8 5. A quadratic function is shown in the table below. Which of the following statements is not true about the function based on this table? Explain your choice. x f x (1) The function has an x intercept of 3. () The function has a y-intercept of 3. (3) The function s leading coefficient is negative. (4) The function has a vertex of 1, 4 1 0 0 3 1 4 3 3 0 4 5 5 1
f x x x 8. 6. Consider the quadratic function whose equation is (a) Sketch a graph of f on the grid provided. y (b) Over what interval is f decreasing? (c) Over what interval is f x 0? x (d) State the range of f. APPLICATIONS 7. The number of meters above the ground, h, of a projectile fired at an initial velocity of 86 meters per second and at an initial height of 6. meters is given by ht 4.9t 86t 6., where t represents the time, in seconds, since the projectile was fired. If the projectile hits its peak height at t 8.775 seconds, which of the following is closest to its greatest height? (1) 65 meters (3) 4 meters () 384 meters (4) 578 meters 8. Physics students were modeling the height of a ball once it was dropped from the roof of a 5 story building. The students found that the height in feet, h, of the ball above the ground as a function of the number of seconds, t, since it was dropped was given by ht 300 16t. From what height was the ball dropped? To the nearest tenth of a second, determine the time at which the ball hits the ground. Provide evidence from a table to support your answer or solve this algebraically if you recall how to.
Lesson 1 Homework Key: 1. (a) 16, 0 (b) 6, -10 (c) 6, 14. (4) 3. (3) 4. () hint: intercepts must be equidistant from the vertex 5. (3) hint: since the parabola has a minimum point at its vertex, it must point UP 6. (a) Graph below (b) x < 1 (c) 4 < x < (d) y 9 or [ 9, ) 7. () 8. 300ft., t=4.3 seconds
LESSON FACTORING WITH GCF AND DOTS In the study of algebra there are certain skills that are called gateway skills because without them a student simply cannot enter into many more complex and interesting problems. Perhaps the most important gateway skill is that of factoring. The definition of factor, in two forms, is given below. FACTOR TWO IMPORTANT MEANINGS (1) Factor (verb) To rewrite a quantity as an equivalent product. () Factor (noun) Any individual component of a product. Always keep in mind that when we factor (verb) a quantity, we are simply rewriting it in an different form that is completely equal to the original quantity. For example, and 3 are factors of 6. It might look different, but 3 is still the number 6 just in a different form. Exercise #1: Rewrite each of the following binomials as a product of an integer with a different binomial. (a) 5x 10 (b) x 6 (c) 6x 15 (d) 6 14x The above type of factoring is often referred to as factoring out the greatest common factor (GCF). This greatest common factor can be comprised of numbers, variables, or both. Exercise #: Write each of the following binomials as the product of the binomial s GCF and another binomial. (a) 3x 6x (b) 0x 5 x (c) 10x 5 x (d) 30x 0 Exercise #3: Rewritten in factored form 0x 36 x is equivalent to (1) x10x 15 (3) 5x4x 7 () 4x5x 9 (4) 9x(x 4)
Trinomials can also sometimes be factored into the product of a GCF and another trinomial. Exercise #4: Rewrite each of the following trinomials as the product of its GCF and another trinomial. (a) x 8x 10 3 3 (b) 10x 0x 5 (c) 8x 1x 0x (d) 6x 15x 1x Another type of factoring that you should be familiar with is in the form of the difference of perfect squares. Once factored, these binomial are known as conjugate pairs: two binomials which have the same terms, but different signs. DIFFERENCE OF TWO SQUARES PATTERN x ax a x a Exercise #5: Write each of the following binomials as the product of a conjugate pair. (a) x 9 (b) 4 x (c) 4x 5 (d) 16 81x Exercise #6: Write each of the following binomials as the product of a conjugate pair. (a) x 1 4 (b) 5 1 9 x (c) 4 81 x 49 9 (d) 36x 49y Factoring an expression until it cannot be factored anymore is known as complete factoring. Complete factoring is an important skill to master in order to solve a variety of problems. When completely factoring an expression, the first type of factoring always to consider is that of factoring out the GCF. Then, see what can be done with the remaining piece. Exercise #7: Using a combination of GCF and difference of perfect squares factoring, write each of the following in its completely factored form. (a) 5x 0 (b) 8x 7 (c) 40 50x (d) 3 3x 48 x
LESSON HOMEWORK: FACTORING WITH GCF AND DOTS FLUENCY 1. Rewrite each of the following binomials as the product of an integer with a different binomial. (a) 10x 55 (b) 4x 40 (c) 6x 45 (d) 18x 9. Rewrite each of the following binomials as the product of its GCF along with another binomial. (a) x 8 x (b) 6x 7 (c) 30x 35x (d) 4x 0x 3 3. Rewrite each of the following binomials as the product of a conjugate pair. (a) x 11 (b) 64 x (c) 4x 1 (d) 5x 9 1 4. Rewrite each of the following trinomials as the product of its GCF and another trinomial. (a) 4x 1x 8 (b) 6x 4x 10 3 (c) 14x 35x 7x (d) 3 0 5 15 x x x 5. Completely factor each of the following binomials using a combination of GCF factoring and conjugate pairs. (a) 6x 150 (b) 36 4x (c) 8x 7 (d) 3 7x 1 x (e) 80 15x (f) 3 x 00 x (g) 8x 51 (h) 44x 99x 3
6. When completely factored, the expression (1) 316 x16 x (3) 3x4x 4 () 3x16 x 16 (4) 34 x4 x 48 3x is written as 7. Which of the following represents the greatest common factor of the terms (1) 36xy (3) () 3 4x y (4) 5 xy x y 6 5 4x y and 18xy? 8. Which of the following is not a factor of 6x 18 x? (1) x 3 (3) 1 () (4) x 9. Which of the following prime numbers is not a factor of the integer 330? (1) 11 (3) 3 () 7 (4) 5 APPLICATIONS 10. The area of any rectangular shape is given by the product of its width and length. If the area of a particular rectangular garden is given by A 15x 35x and its width is given by 5x, then find an expression for the garden s length. Justify your response.
3 11. The volume of a particular rectangular box is given by the equation V 50x x. The height and length of the box are shown on the diagram below. Find the width of the box in terms of x. Recall that V LW H for a rectangular box. x x 5? 1. A projectile is fired from ground level such that its height, h, as a function of time, t, is given by Written in factored form this equation is equivalent to h 16t 80t. (1) h 16t t 4 (3) h 16t t 5 () h 8t t 7 (4) h 8t t 5
1. Lesson Homework Key:. 3. 4. 5. 6. (4) 7. (3) 8. (3) 9. () 10. L = 3x 7 11. L = 5 x 1. (3)
LESSON 3: FACTORING TRINOMIALS Factoring trinomials, expressions of the form ax bx c, is an important skill. Trinomials can be factored if they are the product of two binomials. The two main keys to factoring trinomials are: (1) the ability to quickly and accurately multiply binomials (FOIL) and () the ability to work with signed numbers. We practice both of these skills with four warm-up multiplication problems in Exercise #1. Exercise #1: Without using your calculator, write each of the following products in simplest (a) 3x5x 7 (b) x3x 5 ax bx c form. (c) 5x4x (d) 4x33x 8 Recall from Algebra I: Case One Factoring (a = 1) When factoring trinomials, we are looking for factors of the a c product, which add to the b term. Carefully consider signs when choosing factors. Spend the time to check your choices before moving forward. Using the factors found, re-write the trinomial as the product of two binomials. Exercise #: Factor each trinomial. Show all steps in the decision of factors. (a) x x 35 (b) x 11x 4 (c) x 13x (d) x 5x 50
Recall from Algebra I: Case Two Factoring (a > 1) When factoring trinomials, we are looking for factors of the a c product, which add to the b term. Carefully consider signs when choosing factors. Spend the time to check your choices before moving forward. Re-write the trinomial as 4 terms by splitting the middle term Factor by grouping the polynomial into two groups of two Rearrange by reverse distribution two re-write the trinomial as the product of two binomials. Exercise #3: Factor each trinomial. Show all steps. (a) 3x 19x 40 (b) x 15x 18 (c) 15x 13x (d) 10x 13x 30 (e) 1x 8x 15 (f) 36x 35x 6
LESSON 3 HOMEWORK: FACTORING TRINOMIALS FLUENCY 1. Multiply each of the following binomial pairs and express your answer in simplest trinomial form. (a) x53x (b) 3x85x 1 (c) 8x3x 7 (d) 7x55x. Which of the following is the correct factorization of the trinomial 1x 3x 10? Hint eliminate two of the choices because they are unintelligent guesses based on the leaded term or ending term. (1) 6x13x 10 (3) 4x53x () 6xx 5 (4) 4x53x 3. Written in factored form x 16x 36 is equivalent to (1) x3x 1 (3) xx 18 () x6x 6 (4) x9x 4 4. Write each of the following trinomials in its factored form. Check by multiplying. (a) x 7x 18 (b) x 14x 4 (c) x 17x 30 (d) x 5x 6
(e) x 5x 6 (f) x 15x 44 (g) x 1x 0 (h) x 6x 16 5. Factor each of the following with a leading coefficient other than 1. Show all steps. Check by multiplying. (a) 5x 41x 8 (b) 3x 4x 0 (c) x 9x 15 (d) 7x 39x 0 (e) 18x 5x 8 (f) 0x 11x 4
LESSON 3 HOMEWORK KEY 1. (a) 6x + 11x 10 (b) 15x 43x + 8 (c) 8x + 59x + 1 (d) 35x 11x 10. (4) 3. (3) 4. 5. (a) (5x 1)(x 8) (b) (3x + 10)(x ) (c) (x + 1)(x 15) (d) (7x + 4)(x + 5) (e) (9x 8)(x 1) (f) (5x + 6)(4x 7)
LESSON 4: COMPLETE FACTORING Each expression that we have factored has been the product of two quantities. But, factoring can produce many more than just two factors. In Exercise #1, we first warm-up by multiplying three factors together. Exercise #1: Write each of these in their simplest form. The last two should take little time to do. (a) x4x 7 (b) 5x5x 3 (c) 3x5x 5 (d) 4x3x 3x RECALL: To completely factor an expression means to write it as a product which includes binomials that contain no greatest common factors (GCF s). Exercise #: Consider the trinomial x 4x 6. (a) Verify that both of the following products are correct factorizations of this trinomial. x6x1 xx3 (b) Why are neither of these completely factored? (c) Write each of these in completely factored form by factoring out the GCF of each unfactored binomial. (d) What is true of both complete factorizations you found in part (c)?
In practicality, it is always easiest to completely factor by ALWAYS looking for a GCF first. Once removed, the factoring then either consists of the difference of perfect squares or standard trinomial techniques. Exercise #3: Write each of the following in its completely factored form. (a) 4x 1x 40 (b) 6x 4 (c) x 0x 50 (d) 75 3x (e) 10x 55x 105 (f) 1x 57x 15 (g) 4x 5x 6 (h) 6x 13x 6 (i) 1x 9x 8
LESSON 4 HOMEWORK: COMPLETE FACTORING FLUENCY 1. Find each of the following products in their simplest ax bx c form. (a) 5x6x (b) 3x1x 1 (c) xx 4x 10 Write each of the following expressions in their completely factored form. Always think! () x 14x 36 (3) 5x 70x 45 (4) 3x 19 (5) 3 6 36 96 x x x (6) 8x 7 3 x (7) 8x 1x 8 3 (8) 15x 110x 10 (9) 10x 6x 1x
(10) 8x 67x 4 (11) 1x 0x 3 (1) 18x 39x 15 (13) 45x 0x 3 3 (14) 8x 30x 8 (15) 90x 90x 0x (16) 7x 3 (17) 0x 11x 48
LESSON 4 HOMEWORK KEY 1. (a) 5x 40x + 60 (b) 1x 3 (c) x 3 + 8x + 80x ALWAYS THINK GCF FIRST!. (x 9)(x + ) 3. 5(x + 7)(x + 7) 4. 3(x 8)(x + 8) 5. 6x(x + 8)(x ) 6. 7x( x)( + x) 7. 4(x 1)(x + ) 8. 5(3x 4)(x 6) 9. x(5x + )(x 3) 10. (8x + 3)(x + 8) 11. (6x 1)(x 3) 1. 3(3x + 1)(x 5) 13. 5x(3 x)(3 + x) 14. (4x + 7)(x + ) 15. 10x(3x 1)(3x ) 16. 3(3x 1)(3x + 1) 17. 4(5x )(x + 6)
LESSON 5: FACTORING BY GROUPING You now have essentially three types of factoring: (1) greatest common factor, () difference of perfect squares, and (3) trinomials. We can combine gcf factoring with the other two to completely factor quadratic expressions. Today we will introduce a new type of factoring known as factoring by grouping. This technique requires you to see structure in expressions. It is often used on polynomials of higher power. This is similar to what we do with Case II Trinomials Exercise #1: Factor a binomial common factor out of each of the following expressions. Write your final expression as the product of two binomials. (a) xx 1 7x 1 (b) 5xx 4x (c) x 5x 7 x 7x 1 (d) x 8x 4 x x 4 Exercise #: Write the expression x 3x 4 5x 3 equivalency with x. as the equivalent product of binomials. Test this
When we factor by grouping we first extract common factors from pairs of binomials in four-term polynomials. If we are lucky, we are left with another binomial common factor. Exercise #4: Use the method of factoring by grouping to completely factor the following expressions. 3 3 (a) 3x x 7x 18 (b) 18x 9x x 1 (c) x 5 4x 3 + x 8 3 (d) 5x 10x 0x 40 Exercise #5: Consider the expression x ab ax bx. (a) How can you rewrite the expression so that the first two terms share a common factor (other than 1)? (b) Write this expression as an equivalent product of binomials. Be careful when you use factoring by grouping. Don't force the method when it does not apply. This can lead to errors. 3 Exercise #6: Consider the expression x 10x 7x 1. Explain the error made in factoring it. How can you tell that the factoring is incorrect? x 7 x 5 x 3 x 7x 8 3 x 10x 7x 1 x x 5 7 x 3
LESSON 5 HOMEWORK: FACTORING BY GROUPING FLUENCY 1. Rewrite each of the following as the product of binomials. Be especially careful on the manipulations that involve subtraction. (a) xx 5 7x 5 (b) 4xx 3x (c) x 10 x 3 x 5x 3 (d) x 7x 4 x 4x (e) 4x 3x 1 x x 1 (f) 3x 7x 5 x 5x 4. Max tries to simplify the expression 5x x 3 x 3x 3 as follows: 5x x 3 x 3 x 3 x 35x x 3 x 33x 1 Explain Max s error to him and show him the correct simplification with all steps. 3. Factor each of the following quadratic expressions completely using the method of grouping: (a) 10x 6x 35x 1 (b) 1x 3x 0x 5
4. Factor each of the following cubic expressions completely. 3 3 (a) 5x x 0x 8 (b) 18x 7x x 3 (c) 3 x x x 3 5 50 (d) 8x 10x 1x 15 5. Factor each of the following expressions. Rearrange the expressions as needed to produce binomial pairs with common factors. (a) x ac cx ax (b) xy ab ay bx REASONING 3 6. Consider the expression: x 5x 9x 45. Enter this expression on your calculator and list all of its zeroes (x-values where y = 0). Then, factor it completely. Do you see the relationship between the factors and the zeroes? Explain.
LESSON 5 HOMEWORK KEY 1. (a) (x + 5)(x + 7) (b) (x )(4x 3) (c) (x 3)(x + 15) (d) (x + 4)(3x 5) (e) (x 1)(3x + 1) (f) (x + 5)(x + 11). (x + 3)(3x + 5)Max forgot to distribute the subtraction. 3. (a) (5x + 3)(x + 7) (b) (4x + 1)(3x 5) 4. (a) (5x + )(x )(x + ) (b) (x 3)(3x + 1)(3x 1) (c) (x + )(x + 5)(x 5) (d) (4x + 5)(x + 3) 5. (a) (x c)(x + a) (b) (y + b)(x + a) 6. Zeros at 3, 3, 5 (x 5)(x 3)(x + 3) - if these were solved, it would be the same zeros.
LESSON 6: FACTORING SPECIAL CASES There are certain types of polynomials that can be factored using short-cuts and patterns. The first type is PERFECT SQUARE TRINOMIALS. While all factorable trinomials can be factored using the techniques reviewed earlier in this chapter, there is one special type that has its own trick to it. If you can identify the pattern, it will save you some time. Be extra careful with signs! PERFECT SQUARE TRINOMIAL FACTORING a + ab + b = (a + b) a ab + b = (a b) Recognizing the pattern is the key to using this shortcut. If you recognize that the first term and last term are both perfect squares, check the middle term: it should be DOUBLE the square roots of those terms. For example: x + 1x + 36 You should quickly recognize that x and 36 are both perfect squares Check to see if the pattern works: x = x 36 = 6 (x)(6) = 1x It works! Factor quickly, being thoughtful of the sign used. (x + 6) Exercise #1: Check to see if each of the following fits the Perfect Square Trinomial pattern to factor. If it does not, factor using other methods. (a) y 0y + 100 (b) a + 14a + 49 (c) x + 13x + 36 (d) 16t 40t + 5 (e) 4x + 0x + 9 (f) 9x 4xy + 16y
The second type is SUM or DIFFERENCE OF PERFECT CUBES. For these types, this is the only form of factoring we will consider in this chapter. Be extra careful with signs! SUM OR DIFFERENCE OF PERFECT CUBES a 3 + b 3 = (a + b)(a ab + b ) a 3 b 3 = (a b)(a + ab + b ) same signs always positive First, let s recall some commonly seen cubes: opposite signs (1) 3 = () 3 = (3) 3 = (4) 3 = (5) 3 = (6) 3 = (7) 3 = (8) 3 = (9) 3 = (10) 3 = For example: (8x 3 + 7) You should quickly recognize that 8x 3 and 7 are both perfect cubes and this is SUM Identify the a and b terms 8x 3 = (x) 3 7 = (3) 3 Apply the pattern, ensure proper signs are used throughout. a 3 + b 3 = (a + b)(a ab + b ) 8x 3 + 7 = Exercise #: Factor using sum or difference of perfect cubes. (a) t 3 + 1 (b) 64x 3 + y 3 (c) 7m 3 8 (d) 7x 3 64y 3 (e) x 3 + 8y 3 (f) 15g 3 h 3
FLUENCY LESSON 6 HOMEWORK: FACTORING SPECIAL CASES 1. Factor each of the following using Perfect Square Trinomial factoring. (a) a + 4a + 4 (b) p + p + 1 (c) k + k + 11 (d) 5k 0k + 4 (e) 4x + 1x + 9 (f) 100a 140ab + 49b (g) 49x + 8xy + 4y (h) y 8y + 16
. Factor each of the following using Sum or Difference of Cubes. Write the pattern equation each time before plugging in. (a) a 3 1 (b) x 3 8 (c) p 3 +q 3 (d) k 3 f 3 (e) y 3 8x 3 (f) 7x 3 1 (g) 15t 3 + 8r 3 (h) 16z 3 7w 3
LESSON 6 HOMEWORK: FACTORING SPECIAL CASES 1.. a) (a + ) b) (p + 1) c) (k + 11) d) (5k + ) e) (x + 3) f) (10a + 7b) g) (7x + y) h) (y + 4) a) (a 1)(a + a + 1) b) (x )(x + x + 4) c) (p + q)(p pq + q ) d) (k f)(k + kf + f ) e) (y x)(y + xy + 4x ) f) (3x 1)(9x + 3x + 1) g) (5t + r)(5t 10tr + 4r ) h) (6z 3w)(36z + 18zw + 9w )