Pre-AP Algebra 2 Unit 3 Lesson 1 Quadratic Functions

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1 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 Pass out new Little Black Book Activity 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 = x 2, 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) = x 2 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 2, f mn 2 Standard Form: f(x) = ax 2 + 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 2a, f b 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 = x 2 ; if 0< a < 1, the parabola is flatter than y = x 2 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.

2 Lesson #3-1: Do Now Graphing Parabolas 1. Fill in the table for ( ) and ( ). Then, graph each function on the axes ( ) ( ) Compare ( ) and ( ) Vertex ( ) ( ) Axis of Symmetry Domain Range 2. Given the function f (x) 2x 3 2 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.

3 Lesson #3-1: Do Now 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: Range: How do you find the intercepts and intercept?

4 Lesson #3-1: Do Now 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

5 Lesson #3-1: Homework 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 = x 2 (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. 1) f (x) 3x ) f (x) 2(x 5)(x 4)

6 Lesson #3-1: Homework 3) f (x) x 2 3x ) f (x) 2x 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.

7 Lesson 3-1 Notes Name: Concepts Examples Background Information Graph each function: Parts of a parabola: There are three forms a quadratic equation can be written in: 1) f (x) 2x Vertex Form: Info: Vertex: x-intercepts: Number and types of x-intercepts: Intercept Form: Info: Vertex: x-intercepts: Standard Form: Vertex: x-intercepts: How to find the x-intercepts (roots):

8 Lesson 3-1 Notes Concepts Shape of the Parabola (for all forms): 2) f (x) (x 3)(x 6) Name: Examples 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 x2 3x 4