Unit 6 Part I Quadratic Functions 2/9/2017 2/23/2017 By DeviantArt user MagicFiretrucks Name: By the end of this unit, you will be able to Analyze the characteristics of graphs of quadratic functions Graph quadratic functions in standard form, vertex form, and intercept (factored) form Apply translations to quadratic functions Apply dilations and reflections to quadratic functions Solve quadratic equations by factoring Estimate solutions of quadratic equations by graphing
Table of Contents Key Features of Quadratic Functions... 3 Graphing Parabolas in Standard Form... 5 Solving Equations by Graphing... 6 Vertex Form... 7 Intercept Form... 9 Parabola Transformations... 11 Parabola Transformations Summary... 13
Key Features of Quadratic Functions Warm Up: Complete the table of values and graph the function y = x 2 at right. x -3-2 -1 0 1 2 3 y Key Features: Parabola Standard Form Axis of Symmetry How to find it: Vertex How to find it: Minimum Maximum Domain How to find it: Range How to find it:
Identify Characteristics from Graphs For each graph, identify the axis of symmetry, vertex, and y-intercept. Max/min? (circle one) Y-Intercept: Max/min? (circle one) Y-Intercept: Max/min? (circle one) Y-Intercept: Max/min? (circle one) Y-Intercept: Identify Characteristics from Functions For each equation, identify the axis of symmetry, vertex, and y-intercept. 1. y = 2x 2 + 4x 3 Max/min? 3. y = 3x 2 + 6x 5 Max/min? 2. y = x 2 + 6x + 4 Max/min? 4. y = 2x 2 + 2x + 2 Max/min?
Graphing Parabolas in Standard Form Warm Up: Given the equation y = x 2 + 4x + 3, find the Max/min? How to Graph Quadratic Functions in Standard Form: 1. Find the. Graph it using a. 2. Find and plot the. 3. Find and plot the. 4. Plot more. 5. Draw a. Graph the following quadratic functions. 1. Graph f(x) = 2x 2 + 2x 1 2. Graph f(x) = 3x 2 6x + 2 Other points: Other points:
3. f(x) = x 2 + 6x 2 Other points: 4. The cheerleaders at Oswego High School want to use a T-shirt cannon to launch T-shirts into the crowd every time the Panthers score a touchdown. The height of the T-shirt can be modeled by the following equation y = 16x 2 + 48x + 6. Graph the function. At what height will the T-shirts be launched from? What will the maximum height of the T-shirts be? How long will it take the shirts to reach their maximum height?? Solving Equations by Graphing The solutions are the same as the x-intercepts of the graph. This is because x-intercepts occur where y=0. What does a quadratic function look like when it has 0 solutions, 1 solution, and 2 solutions?
Vertex Form Warm Up: As Ms. Abels moves the h slider, what happens? As Ms. Abels moves the k slider, what happens? A quadratic function in vertex form looks like this: f(x) = a(x h) 2 + k How to Graph Quadratic Functions in Vertex Form: 1. Find the at the point. 2. Draw the. 3. Find and plot the. 4. Find and plot the. 5. Plot more. 6. Draw a. Examples: Graph each function. 1. f(x) = (x 1) 2 3 vertex: axis of symmetry: other points:
2. f(x) = 2(x + 2) 2 + 3 vertex: axis of symmetry: other points: 3. f(x) = 3(x + 1) 2 5 vertex: axis of symmetry: other points: 4. f(x) = (x 2) 2 + 1 vertex: axis of symmetry: other points:
Intercept Form Warm Up: A quadratic function in intercept form (also known as factored form) looks like this: f(x) = a(x p)(x q) How to Graph Quadratic Functions in Vertex Form: 1. Plot the at the points and. 2. Draw the. 3. Find and plot the. 4. Find and plot the. 5. Plot more. 6. Draw a. Examples: Graph each function. 1. f(x) = (x 7)(x 3) x-intercepts: axis of symmetry: vertex:
2. f(x) = 2(x + 3)(x 1) x-intercepts: axis of symmetry: vertex: 3. f(x) = (x + 2)(x 4) x-intercepts: axis of symmetry: vertex: 4. f(x) = 1 (x + 2)(x 4) 2 x-intercepts: axis of symmetry: vertex:
Parabola Transformations Warm Up: For quadratic functions, the parent function is simply f(x) = x 2. All other quadratic functions are variations of this parent function. Different transformations can change the position or size of this graph. Graph the parent function at right. Parent Function: f(x) = x 2 Transformation #1: On your graphing calculator, graph the following three functions. Sketch the graph at right, labeling each graph. 1. f(x) = x 2 2. f(x) = x 2 + 3 3. f(x) = x 2 5 Describe the transformation that you see in your own words. Describe how the graph of each function below is related to the graph of f(x) = x 2. 1. f(x) = x 2 7 2. g(x) = 5 + x 2 3. h(x) = 5 + x 2 4. f(x) = x 2 + 1 Transformation #2: On your graphing calculator, graph the following three functions. Sketch the graph at right, labeling each graph. 1. f(x) = x 2 2. f(x) = (x 5) 2 3. f(x) = (x + 7) 2 Describe the transformation that you see in your own words. Describe how the graph of each function below is related to the graph of f(x) = x 2. 1. f(x) = (x 3) 2 2. g(x) = (x + 2) 2 3. h(x) = (x 1) 2 4. f(x) = (x + 10) 2
Transformation #3: On your graphing calculator, graph the following functions. Sketch the graph at right, labeling each graph. 1. f(x) = x 2 2. f(x) = x 2 Describe the transformation that you see in your own words. Describe how the graph of each function below is related to the graph of f(x) = x 2. 1. f(x) = x 2 3 2. g(x) = (x + 2) 2 3. h(x) = (x 1) 2 + 4 4. f(x) = (x + 10) 2 2 Transformation #4: On your graphing calculator, graph the following three functions. Sketch the graph at right, labeling each graph. 1. f(x) = x 2 2. f(x) = 1 2 x2 3. f(x) = 3x 2 Describe the transformation that you see in your own words. Describe how the graph of each function below is related to the graph of f(x) = x 2. 1. f(x) = 2x 2 3. h(x) = 1 3 x2 + 2 2. g(x) = 5x 2 2 4. f(x) = 2(x 1) 2 2
Parabola Transformations Summary f(x) = a(x h) 2 + k Look familiar? That s because this is vertex form. Order Matters: Stretch/compress/reflect must come before horizontal/vertical translation. Identifying Transformations Describe how the graph of each function is related to the graph of f(x) = x 2. 1. g(x) = x 2 11 2. h(x) = 1 (x 2 2)2 3. h(x) = x 2 + 8 4. g(x) = x 2 + 6 5. g(x) = 4(x + 3) 2 6. h(x) = x 2 2 Identifying Equations from Graphs Match each equation to its graph. 1. y = 1 3 x2 4 2. y = 1 (x + 3 4)2 4 3. y = 1 3 x2 + 4 4. y = 3x 2 2 5. y = x 2 + 2 6. y = 2(x + 3) 2 2