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pg 1 4.1 Quadratic Functions and Transformations A: Quadratic Functions A QUADRATIC function is an equation in the form: y ax 2 bx c The _________ of a quadratic equation is a PARABOLA. Ex. Circle the parabolas. Cross out the others. y 2 x 2 15 x 18 y 6 x 9 y x2 4x 4 y 16 x 22 B: Translations of x2 All parabolas are a translation of the parent graph y x 2 We discussed translations in chapter 2, lesson 2.6. Use your graphing calculator to graph the following translations and state the transformations to the parent function. Ex. 2. y 3( x 1) 2 5 Ex. 1. y 2( x 3) 2 1 pg 2 Ex. 3. Match the graph with the equation A B C D C: Vertex Form Quadratic functions can also be written in “vertex form”: y a ( x h) 2 k The Vertex = ____________ Domain: the set of all x values for the equation For an up/down parabola, usually: The Axis of Symmetry = ____________ ______________________________ “a” = _________________ of opening Range: positive: opens _______ the set of all y values for the equation For an up/down parabola, usually: negative: opens _______ _______________________________ Min/Max: The highest or lowest “y” value on the graph, the y coordinate of the ___________ Ex. 4 Use your graphing calculator to graph: y 2( x 2)2 5 State: Vertex: __________ Min or Max? ______ Value:_____ A.O.S.: _________ Domain: _________________ Range: ___________________ pg 3 Ex. 5 Use your graphing calculator to graph: y 3x 2 4 State: Vertex: __________ Min or Max? ______ Value:_____ A.O.S.: _________ Domain: _________________ Range: ___________________ WITHOUT A GRAPHING CALCULATOR: Use a t-chart to graph the following. State the vertex and the equation of the axis of symmetry. Ex. 6 y ( x 5)2 2 Vertex: _______ Make It Happen: A.O.S.: _______ 1. Determine the vertex. Plot the point. 2. Determine the A.O.S. Draw in the vertical line. 3. Pick a few x-values for your t-chart that are to the right of the A.OS. Ex. 7 y 2( x 4)2 6 4. Use your t-chart to find 2 points on the graph. Vertex: _______ A.O.S.: _______ 5. Use your A.O.S. to reflect these points to the other side. 6. Sketch in the curve. pg 4 D: Using Vertex Form Find an Equation from a graph Recall that vertex form: y a( x h)2 k (h,k) is the vertex (x,y) is any point on the curve Example 8: Find the equation that models the graph at the right. Step 1: What is the VERTEX: ______ so h = _____ and k = ______ Step 2: What is a POINT on the graph? _______ so x = _____ and y = _______ Step 3: Substitute into vertex form and solve for a. y a ( x h) 2 k Step 4: Plug a, h, and k back into your vertex form. Example 9: Find the equation that models the graph Use these to FIND: ____ pg 5 4.2 Standard Form of a Quadratic Function A: Standard Form, Finding a Vertex If a quadratic function is ALREADY in vertex form y a( x h)2 k , you can find the vertex just by looking at the equation. If a quadratic function is in STANDARD FORM y ax 2 bx c , you have to do a little work first to find the vertex. Ex. 1 Use a graphing calculator to graph y x2 2 x 1 Vertex: ______ Min/Max? _______ Value: ______ Ex. y x2 2 x 1 A.O.S.: ______ Domain: ________________________________ Finding a vertex Range: _________________________________ pg 6 Find the vertex, the axis of symmetry, the minimum or maximum value, the domain, and the range. Ex. 2 y 2 x2 8x 1 Ex. 3 y 3x 2 6 x 9 Ex. 4 WITHOUT A GRAPHING CALCULATOR Use a t-chart to graph y x 2 2 x 3 State the vertex and the axis of symmetry pg 7 B: Standard Form Vertex Form y ax2 bx c y a( x h)2 k Ex. 5 Rewrite y x 2 4 x 5 in vertex form. Step 1: Identify a & b Step 2: Find the x coordinate of the vertex. This value = h Step 3: Substitute in the x-coordinate to find the y-coordinate. This value = k Step 4: Write out vertex form. Step 5: Substitute a, h, and k (Simplify if needed) Rewrite in vertex form Ex. 6 y 2 x 2 10 x 7 Ex. 8 Which is the graph of y 3x 2 4 x 6 ? Ex. 7 y x 2 16 x 66 pg 8 4-3 Modeling with Quadratic Functions A: What is a Model? How do I make one? A model is a way of using math to describe a real world situation. To model a quadratic equation, you need to know the values of a, b, and c in y ax 2 bx c . Ex. 1 What is the equation of the parabola that contains the points (0,0), (1,-2), and (-1,-4)? Make a Plan We need a, b, c We HAVE three points (x, y) Make it Happen Substitute each of the three points (x,y) into y ax 2 bx c Solve the 3x3 systems of equations (like section 3.5) Substitute a, b, & c into y ax 2 bx c y ax 2 bx c Use (0,0) Use (1,-2) Use (-1,-4) pg 9 X 0 2 1 Y 3 5 6 Ex. 2 Write the equation of the quadratic equation that passes through the points on the table. B: Quadratic Regression Sometimes your data points will not be EXACTLY a parabola, but the basic shape of the data is parabolic. We can use a graphing calculator to ESTIMATE the equation of a parabola that models the data. Ex. 3 What is the quadratic model for the data? (start by changing to a 24 hr clock!) Quadratic Regression in Graphing Calculator 1. Press the STAT key 2. Choose Option 1 “EDIT” 3. Enter your “x” data in the L1 list Enter your “y” data in the L2 list 4. Press the STAT key 5. Arrow right to the CALC menu and choose #5 Quad Reg 6. Enter 7. Use the estimated a, b, &c to write your equation pg 10 Ex. 4 Use quadratic regression to find a quadratic model for the data in the chart. X represents the number of years since 1985. a = _________________________ x=0 b = ________________________ x=5 x=15 c = ________________________ x=19 Equation: _______________________________________________ Use the equation to estimate the number of subscribers in 1995. Ex. 5 Use quadratic regression to find a quadratic model for the data in the chart. x represents the number of years since 1976 a = _____________________ x=0 b = _____________________ x=10 x=20 c = _____________________ Equation: ___________________________________________________ Use the equation to estimate the price per gallon in 2006 (x = 30). x=29 pg 11 The main goal of Lessons 4.4-4.7 is to ____________ quadratic equations. This means we are looking for values of x ( _____________________) that make the equation ax 2 bx c 0 true. y For example: The solutions to x 2 x 6 0 are x 2 and x 3 . Look at the graph: y x 2 x 6 What do you notice about x 2 and x 3 ? Plug these values in to confirm. 4.4 FACTORING What is factoring? A very useful algebra skill that we will use in NEARLY EVERY CHAPTER FOR THE REST OF THIS COURSE A way to “undo” the distributive property or the “FOIL” process A method of taking a quadratic expression like ax 2 bx c and writing it as a PRODUCT of two FACTORS LOOK FOR _______________________ ! In this section we will REVIEW several factoring techniques: 1. 2. 3. 4. Greatest Common Factor x 2 bx c ax 2 bx c when a 1 Difference of Two Perfect Squares a 2 b 2 pg 12 A: Greatest Common Factor Recall Distributive Property Multiply The first step in any factoring problem is to look for the Greatest Common Factor for all of your terms and “undistribute”. 2 x(3x 5) Your final answer will look like the GCF (leftovers). Examples: FACTOR! A1. 7 x 2 21 A2. 11x 2 33 x A3. 4 x3 x 2 16 x A4. 3x 2 3x A5. 9 x 3 45 x 2 A6. 12 x 2 15 B: Trinomials ax 2 bx c a 1 After you’ve checked for a Greatest Common Factor. Check to see if you have a trinomial with an “a”= 1 or “a” = -1. You should try to find two special numbers that have a Recall “FOIL” Process Multiply ( x 2)( x 3) PRODUCT of “c” mxn=c SUM of “b” m+n=b Factors: (x + m) ( x + n) Examples: FACTOR! B1. x 2 9 x 20 B2. x 2 14 x 72 B3. x 2 17 x 30 pg 13 B4. x 2 13x 12 C: Trinomials ax 2 bx c a 1 Recall “FOIL” Process Multiply (2 x 3)( x 5) B5. x 2 14 x 32 x 2 11x 30 B6. After you’ve checked for a Greatest Common Factor. After you’ve checked for a Trinomial with a = 1 or a = -1. Look to see if you have a Trinomial with a = some other number. You are going to have to use guess and check to “DE-FOIL” ax 2 bx c (hx m)( gx n) a h g b hn gm c m n the FIRST Terms the OUTSIDE + INSIDE terms the LAST terms Examples: FACTOR! C1. 2 x 2 11x 12 C2. 4x2 4 x 3 C3. 2 x2 7 x 6 C4. 5x2 7 x 6 pg 14 D: Perfect Square Trinomials Look for a pattern like this: Multiply: (2 x 3) 2 a 2 2ab b2 (a b)2 a 2 2ab b2 (a b)2 Essentially you can do these like part (C), but if you recognize that the first term and the last term are perfect squares, it can make the problem much easier to factor! Examples: Factor! D1. 4 x 2 24 x 36 E: Difference of Two Perfect Squares Multiply: ( x 6)( x 6) D2. 64 x 2 16 x 1 Look for a pattern like this: a 2 b2 (a b)(a b) Essentially you can do these like part (C), but if you recognize that the first term and the last term are perfect squares, it can make the problem much easier to factor! Also be careful, unless there is a GCF Examples: Factor! E1. x 2 81 E2. x2 1 E3. 9x2 4 E4. 16 x 2 100 y 2 a 2 b 2 does not factor! pg 15 F: Factor Completely 1. Check for GCF 2. Factor what’s leftover 3. Check each factored term to see if it will factor again! Examples: Factor Completely F1. 2 x 2 50 F2. 2 x 2 32 x 128 F3. 5 x 2 25 x 70 F4. 3x 2 9 x F5. 2 x 2 22 x 60 F6. 12 x 2 10 x 12 F7. 7 x 2 700 F8. 18 x 2 12 x 2 F9. x 2 12 x 35 pg 16 4.5 Quadratic Equations A: Solve by Graphing Vocab: A solution to a the quadratic equation ax 2 bx c 0 is a value of x that makes the equation true. A zero of a function f ( x) ax 2 bx c is a value of x that makes f(x) = 0. y An x-intercept of the graph of f ( x) ax 2 bx c is the x value on the graph where y f ( x) 0 . This is the place where the graph crosses the x-axis. For real numbers, “Solution” “Zero” and “X-intercepts” are basically synonyms and can be used interchangeably. For the equation y x 2 x 6 whose graph is shown: The solutions to x 2 x 6 0 x The zeros of f ( x) x 2 x 6 The x-intercepts of y x 2 x 6 Finding a “zero” in a graphing calculator Given an equation ax 2 bx c 0 1. Enter y ax 2 bx c 2. Press “Graph” 3. Look at your graph. How many zeros are there? 4. Press 2nd Trace (Calc). Choose option 2 “Zero” 5. The calculator will ask you for the left bound. Decide which zero you are looking for first. Type in an x value to the left of your zero. Press enter. 6. The calculator will ask for a right bound. Type in an x value to the right of your zero. Press enter. 7. The calculator will ask you to guess where you think the zero is. Just press enter again. 8. The calculator tells you the zero of the function. Repeat steps 4-8 for any additional zeros. pg 17 Examples: Use your graphing calculator to find the zeros of the following functions. Give each answer to at most two decimal places. A1. 3x 2 24 x 45 0 A2. 3x 2 2 x 2 0 B: Solve by Factoring You can also find the zeros of a quadratic function algebraically using factoring and the zero product property. Make it Happen! Solve by Factoring! Examples: Solve by Factoring B1. B3. x2 5x 6 0 x 2 2 x 24 0 B2. B4. x 2 7 x 12 4 x2 6 x 0 1. Make sure your equation looks like ax 2 bx c 0 2. Factor completely. 3. Set each factor = 0. 4. Solve for x. B5. 3x 2 45 24 x pg 18 B6. 3x 2 x 4 B7. 25 x 2 20 x 4 0 B8. 9 x 2 49 B9. 3x 2 4 x 32 B10. 10 x 2 3 x 4 B11. 2 x 2 32 0 pg 19 4.6 Completing the Square Quick Review! Simplifying Square Roots A square root essentially asks the question: “What number times itself gives me this value?” If your value is a perfect square, then that number will be an integer. Examples: 49 144 If your value is NOT a perfect square, then you need to simplify the radical using the multiplication property. a b a b Examples: Break your value down into factors, one of which is a perfect square. Also, don’t forget: 100 75 is NOT A REAL NUMBER. A: Solve by Finding Square Roots Examples: Solve A1. 4 x 2 10 46 72 A2. 3 x 2 5 25 ? 2 45 100 is impossible for real numbers! Make it Happen! If your ax 2 bx c 0 equation has no “bx” term, you can solve by taking the square root of both sides of the equation. 1. Use algebra to isolate the x 2 x2 n 2. Take the square root of both sides of the equation. Don’t forget the ! x n 3. Write as two equations. x n and x n pg 20 A3. 2 x 2 9 13 A4. 3 x 2 96 0 B: Perfect Square Trinomials Recall that a trinomial in the form: a 2 2ab b2 (a b)2 and a 2 2ab b2 (a b)2 We can solve this form of equation using square roots as well! Examples: Solve B1. ( x 1)2 49 B2. ( x 8)2 20 B4. 4 x 2 20 x 25 17 Sometimes you will need to factor 1st! B3. x 2 14 x 49 25 pg 21 C: Completing the Square Even if you don’t have a perfect square trinomial, you can still solve using this process using a technique called “completing the square” Here’s How it Works Given a polynomial: Example: Complete the square! x 2 bx C1. 2 Add b 2 Factor b x 2 x 2 10 x C2. x 2 14 x 2 x2 2 x x2 5x C3. C4. Using completing the square to solve quadratic equations Make it happen: Examples: Solve by Completing the Square 1. Isolate the C5. x 2 bx x2 4x 2 0 b 2. Add 2 2 to both sides b 3. Simplify (Factor) x 2 4. Solve using square roots. C6. x 2 10 x 16 0 C7. x 2 18 x 64 0 2 pg 22 D: Vertex Form Completing the square can be useful for converting a standard form parabola into vertex form. D1. y x 2 10 x 9 D2. y x 2 18x 13 pg 23 4.7 The Quadratic Formula The quadratic formula is derived by taking the standard form of the quadratic equation: ax 2 bx c 0 and solving for x using the completing the square process. Example: Solve using the quadratic formula 2 x2 5x 3 0 pg 24 The portion of the equation under the radical is called the discriminant. The discriminant tells us a lot about the types of answers we might get. Examples: a) Find the discriminant. b) Solve using quadratic formula c) How many REAL solutions did you get? Are they rational or irrational? 1. 3x 2 8 x 16 0 2. 4x2 x 2 0 3. 9 x 2 12 x 4 0 4. x2 5x 8 0 pg 25 Remember that graphically, a real solution to a quadratic equation represents an x-intercept of the graph of the equation. Positive Perfect Square: Rational (ex. 3, ½, -4, etc) Not Perfect Square: Irrational (ex. 2 33 , etc.) Zero Negative pg 26 Concept Byte: Writing Equations From Roots Recall: A SOLUTION is a value of x that makes the quadratic equation ax 2 bx c 0 true. Sometimes this is also called a ZERO of a function or a ROOT of an equation. Synonyms SOLUTION = ZERO = ROOT On a graph A “REAL” SOLUTION = ZERO = ROOT 2 x 2 11x 5 0 x 5 x 14 0 ( x 7)( x 2) 0 2 Recall: “SOLVE BY FACTORING” x7 0 x 7 X-Intercept (2 x 1)( x 5) 0 or x2 0 x2 2x 1 0 2 x 1 x x5 0 x 5 1 2 We can work backwards from the roots (the solutions) to find the original equation. Example: Write a quadratic equation with each pair of values as roots. Steps: 1. If “c” is a root, then (x –c ) is a factor. 2. Write your roots as the product of two factors = 0. 1. -7, 2 2. -5, - ½ roots: b, c equation: ( x – b) (x – c) = 0 3. Multiply using FOIL and simplify. 4. Multiply through by a constant to eliminate fractions if needed. pg 27 3. 2, 10 4. 5. 2 , -9 3 6. -5, 5 3 1 , 4 2 Putting it all together 4.4-4.7 4.4 Factoring Patterns: Greatest Common Factor Trinomial Trinomial a = 1 16 x 2 20 x x 2 29 x 100 Trinomial a ≠ 1 3x 2 17 x 10 Difference of Perfect Squares “Factor Completely” x 2 64 x 2 3x 28 25 x 2 4 Perfect Square 25 x 2 30 x 9 20 x 2 80 pg 28 4.5 Solve by Factoring Solve by Factoring Steps: 1. Get ax 2 bx c 0 x 2 144 0 x 2 6 x 40 0 9 x 2 18 x 0 2x2 9 x 4 0 2. Factor 3. Set each factor = 0 4. Solve 4.6 Solve by Completing the Square Solve by Completing the Square Steps: 1. Isolate x 2 bx b 2 2 2. Add to both sides 3. Factor and simplify 4. Solve using square roots x2 4x 5 8 x 2 2 x 35 0 pg 29 4.7 Solve by Quadratic Formula Solve by Quadratic Formula Steps: If Then ax 2 bx c 0 x 2x2 x 4 2 b b2 4ac 2a 8x2 1 7 x Discriminant: Steps: Find b 2 4ac Positive 2 real solutions Zero 1 real solutions Negative 0 real solutions Find the Discriminant. State the number of real solutions. 10 x 2 x 9 0 3 x 2 6a 3 0 9 x 2 6 x 11 0 pg 30 4.8 Complex Numbers A. Imaginary Numbers 25 Consider: Has no REAL solution because (?)2= -25 Imaginary Numbers were created to solve equations that did not have real solutions. The imaginary unit “i” is used to represent: i 1 and i 2 1 Ex. Simplify the following 1. 36 2. 5. 90 6. 9. 2i 3i 10. 100 3. 7 3 8 7. 2 6 7i 3i 2i 11. 5i Numbers like 3i , 97i , and 7i are called PURE IMAGINARY NUMBERS. 2 4. 50 8. 3 10 4 3 12. 3i 3 pg 31 B: COMPLEX NUMBERS A COMPLEX NUMBER has a real part “a” and an imaginary part “bi”. A complex number looks like: a bi Working with complex numbers: Treat i like any other variable… 1. (3 4i) (7 5i) 2. (9 8i) (7 10i) 3. (7 6i ) (6 10i) 4. (17 20i) (12 30i) 5. (5 3i)(4 6i) 6. (8 7i)(2 5i) 7. 7i (9 3i) 8. 4i (20 30i ) 9. (3 5i ) 2 10. (4 2i)(4 2i) 11. (6 5i)(6 5i) Numbers in the form a bi and a bi are called complex conjugates. pg 32 C: Solving Equations Remember that a quadratic equation with no real solutions has a graph that has no x-intercepts. This type of quadratic equation does have SOLUTIONS, though… COMPLEX SOLUTIONS. SOLVE! 1. x2 9 0 4. 3x 2 x 2 0 2. 5 x 2 20 0 3. x 2 15 0 Quadratic Formula for ax 2 bx c 0 b b2 4ac x 2a 5. x2 4x 5 0