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Pre-AP Algebra 2 Unit 3 – Lesson 1 – Quadratic Functions Objectives: The students will be able to Identify and sketch the quadratic parent function Identify characteristics including vertex, axis of symmetry, x-intercept, and y-intercept Translate the quadratic function using a (stretch), h (horizontal) and k (vertical) Identify domain and range of quadratic functions Materials: Little Black Book; Do Now worksheet; pairwork; hw #3-1 Time Activity Pass out new Little Black Book 15 min Do Now Hand out the Graphing Parabolas sheet to students. First they compare the parent function for ( ) , to understand that is not the same as slope, but does affect the steepness of the graph. They are given a parabola in vertex form, and must identify the vertex and x-intercepts, which way the parabola is opening and its width relative to the graph of y = x2, and then make a graph. They also have one intercept form parabola. They must make an x-y table to graph it. Review the problems, and discuss how they could use the given intercept form function to easily find the x-intercepts and the vertex (i.e. the midpoint of the x-intercepts). 30 min Direct Instruction Background Information: 1) Identify the parts of a parabola on the given example: vertex, axis of symmetry, x-intercepts (roots) 2) Determine the number of possible roots of a parabola: where does f(x), g(x), and h(x) = 0? 3) To find the roots, set f(x) = 0 and solve. Concepts: There are three forms a quadratic equation can be written in: 1) Vertex Form 2) Intercept Form 3) Standard Form Vertex Form: f(x) = a(x – h)2 + k Comes from translating and transforming the function f(x) = x2 to f(x) = a(x – h)2 + k. x-intercepts: set equal to 0 and solve by working backward vertex: (h, k) Intercept Form: f(x) = a(x – m)(x – n) This can also be called “factored form”. x-intercepts: set each factor equal to 0 and solve. vertex: halfway between the x-intercepts; find the average: mn mn 2 , f 2 Standard Form: f(x) = ax2 + bx + c x-intercepts: set equal to 0, and solve by factoring or using quadratic formula vertex: find average if you factored it; otherwise, use the formula b b , f 2a 2a . For all Forms: If a is positive, the parabola opens up; if a is negative, the parabola opens down If |a| > 1, the parabola is steeper than y = x2; if 0< |a| < 1, the parabola is flatter than y = x2 To Graph Any Parabola: Plot the vertex (and x-intercepts, if they are integers). Draw a dotted axis of symmetry (lightly). Plot the y-intercept f(0) and its reflection. If you need more points, pick x-values on one side of the axis and plug them in to the function. Plot them, and their reflections. Pre-AP Algebra 2 Lesson #3-1: Do Now Name: _______________________ Graphing Parabolas 1. Fill in the table for ( ) and ( ). Then, graph each function on the axes. ( ) ( ) -3 -2 -1 0 1 2 3 Compare ( ) and ( ) ( ) ( ) Vertex Axis of Symmetry Domain Range 2 2. Given the function f (x) 2 x 3 8 , a. Describe the translation and transformation from y x 2 . b. What is the vertex of the graph of f (x) ? c. Does the graph of f (x) open up or down? d. Is it steeper or flatter than the graph of y x 2 ? e. Find the x-intercepts of the graph by solving f (x) 0 . Pre-AP Algebra 2 Lesson #3-1: Do Now Name: _______________________ f. Use the previous work to make a graph of f (x) . g. Compare ( ) to the parent function. ( ) ( ) Vertex Axis of Symmetry Domain Range intercepts intercept ( ) h. Generalize: Given a function in the form ( ) Vertex: Axis of Symmetry: Domain: How do you find the Range: intercepts and intercept? Pre-AP Algebra 2 Lesson #3-1: Do Now Name: _______________________ 3. Given the function ( ) ( )( ), a. Find the x-intercepts by solving f (x) 0 . b. Make an x-y table so that you can graph the parabola. Use the x-intercepts to figure out what x-values to put in the table. Make sure the table goes beyond the x-intercepts. c. Draw the graph of f (x) . d. Which way does the graph open? Is it steeper or flatter than the graph of ? e. Use the graph to determine the vertex of f (x) . f. Explain how you can find the vertex without first graphing the parabola. See problem 4 for a hint if you are stuck. 4. Plot the two numbers on a number line. Then, determine the number exactly halfway between the two numbers. a. 3 and 9 b. 2 and 15 Pre-AP Algebra 2 Lesson #3-1: Homework Name: _______________________ HW #3-1: Graphing Parabolas Practice Check for Understanding Can you complete these problems correctly by yourself For each function, determine the following information: 1. The form the function is written in. 2. Its shape compared to the graph of y = x2 (opens up/opens down; steeper/flatter) 3. The vertex 4. The x-intercepts Then, graph the parabola. Make sure to review your notes before you begin. 2 1) f (x) 3 x 2 75 1. 2. 3. 4. 2) f (x) 2(x 5)(x 4) 1. 2. 3. 4. Pre-AP Algebra 2 Lesson #3-1: Homework Name: _______________________ 3) f (x) x 2 3x 10 1. 2. 3. 4. 4) f (x) 2x 2 8 1. 2. 3. 4. Spiral What do you remember from Algebra 1and our previous units? (these are skills we will need in this unit)Work on a separate sheet a paper 1. Given ( ) and ( ) . Find the following )( ) a. ( d. ( ( )) b. [ ( )] e. ( ( )) c. [ ( )] 2. Factor the following trinomials a. b. c. d. Pre-AP Algebra 2 Lesson 3-1 –Notes Concepts There are three forms a quadratic equation can be written in: Name:________________________ Examples Graph each function: Background Information Parts of a parabola: 2 1) f (x) 2 x 3 2 Vertex Form: Info: Vertex: x-intercepts: Number and types of x-intercepts: Intercept Form: Info: Vertex: x-intercepts: Standard Form: Vertex: How to find the x-intercepts (roots): x-intercepts: Pre-AP Algebra 2 Lesson 3-1 –Notes Name:________________________ Concepts 2) f (x) (x 3)(x 6) Shape of the Parabola (for all forms): To Graph a Parabola: 1. Plot the vertex (and x-intercepts, if they are integers). 2. Draw a dotted axis of symmetry (lightly). 3. Plot the point 0, f 0 (the y-intercept) and its reflection. 4. If you need more points, pick x-values on one side of the axis and plug them in to the function. Plot them, and their reflections. 3) f (x) 1 2 x 3x 4 2 Examples