8 Graphing Quadratic Functions - Big Ideas Learning

1 Graphing Quadratic 8 Functions Graphing f (x) = ax 2. Graphing f (x) = ax 2 + c Graphing f (x) = ax 2 + bx + c Graphing f (x) = a(x h)2 + k Using Intercept Form Comparing Linear, Exponential, and Quadratic Functions SEE the Big Idea Town Population l ion (p. Populati (p 450). 450)). Satellite Dish (p. (p 443). Roller Coaster (p. 434). Firework Explosion (p. (p 423). Garden Gard Ga rden Waterfalls den Water at llss (p. erffall fall (p 416). 416). Mathematical Thinking: Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.)

2 Maintaining Mathematical Proficiency Graphing Linear Equations ( ). Example 1 Graph y = x 1. Step 1 Make a table of values. x y = x 1 y (x, y). 1 y = ( 1) 1 0 ( 1, 0). 0 y = (0) 1 1 (0, 1). 1 y = (1) 1 2 (1, 2). 2 y = (2) 1 3 (2, 3). y Step 2 Plot the ordered pairs. 2. y = x 1. Step 3 Draw a line through the points. ( 1, 0) 2 x (0, 1) (1, 2). Graph the linear equation. 4. (2, 3). 1. y = 2x 3 2. y = 3x + 4. 1. 3. y = 2 x 2 4. y = x + 5. Evaluating Expressions ( ). Example 2 Evaluate 2x2 + 3x 5 when x = 1.

3 2x2 + 3x 5 = 2( 1)2 + 3( 1) 5 Substitute 1 for x. = 2(1) + 3( 1) 5 Evaluate the power. =2 3 5 Multiply. = 6 Subtract. Evaluate the expression when x = 2. 5. 5x 2 9 6. 3x 2 + x 2. 7. x 2 + 4x + 1 8. x 2 + 8x + 5. 9. 2x 2 4x + 3 10. 4x 2 + 2x 6. 11. ABSTRACT REASONING Complete the table. Find a pattern in the differences of consecutive y-values. Use the pattern to write an expression for y when x = 6. x 1 2 3 4 5. y = ax 2. 403. Mathematical Mathematically proficient students use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, Thinking determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

4 ( ). Problem-Solving Strategies Core Concept Trying Special Cases When solving a problem in mathematics, it can be helpful to try special cases of the original problem. For instance, in this chapter, you will learn to graph a Quadratic function of the form f (x) = ax2 + bx + c. The problem-solving strategy used is to first graph Quadratic Functions of the form f (x) = ax2. From there, you progress to other forms of Quadratic Functions . f (x) = ax2 Section f (x) = ax2 +c Section f (x) = ax2 + bx + c Section f (x) = a(x h)2 +k Section Graphing the Parent Quadratic Function Graph the parent Quadratic function y = x2.

5 Then describe its graph. SOLUTION. The function is of the form y = ax 2, where a = 1. By plotting several points, you can see that the graph is U-shaped, as shown. y 10. 8. 6. 4. y = x2. 2. 6 4 2 2 4 6x The graph opens up, and the lowest point is at the origin. Monitoring Progress Graph the Quadratic function. Then describe its graph. 1. y = x2 2. y = 2x2 3. f (x) = 2x 2 + 1 4. f (x) = 2x 2 1. 1 1. 5. f (x) = 2 x 2 + 4x + 3 6. f (x) = 2 x 2 4x + 3 7. y = 2(x + 1)2 + 1 8. y = 2(x 1)2 + 1. 9. How are the graphs in Monitoring Progress Questions 1 8 similar?

6 How are they different? 404 Chapter 8 Graphing Quadratic Functions Graphing f (x) = ax2. Essential Question What are some of the characteristics of the TEXAS ESSENTIAL graph of a Quadratic function of the form f (x) = ax 2? KNOWLEDGE AND SKILLS. Graphing Quadratic Functions Work with a partner. Graph each Quadratic function. Compare each graph to the graph of f (x) = x2. a. g(x) = 3x2 b. g(x) = 5x2. y y 10 4. f(x) = x 2. 8. 6 4 2 2 4 6x 6 4. 4 8. 2 f(x) = x2 12. 16. 6 4 2 2 4 6x 1 2. c. g(x) = d. g(x) =.

7 10. x y y 6. 10. 4. 8. 2 f(x) = x 2. 6. 6 4 2 2 4 6x 4 f(x) = x 2. 2. 2. 4. 6 4 2 2 4 6x 6. REASONING. To be proficient in math, you need to make sense Communicate Your Answer of quantities and their 2. What are some of the characteristics of the graph of a Quadratic function of relationships in the form f (x) = ax2? problem situations. 3. How does the value of a affect the graph of f (x) = ax2? Consider 0 < a < 1, a > 1, 1 < a < 0, and a < 1. Use a Graphing calculator to verify your answers. 4. The figure shows the graph of a Quadratic function 7.

8 Of the form y = ax2. Which of the intervals in Question 3 describes the value of a? Explain your reasoning. 6 6. 1. Section Graphing f(x) = ax 2 405. Lesson What You Will Learn Identify characteristics of Quadratic Functions . Graph and use Quadratic Functions of the form f (x) = ax2. Core Vocabul Vocabulary larry Quadratic function, p. 406 Identifying Characteristics of Quadratic Functions parabola, p. 406. A Quadratic function is a nonlinear function that can be written in the standard form vertex, p. 406.

9 Y = ax2 + bx + c, where a 0. The U-shaped graph of a Quadratic function is called axis of symmetry, p. 406 a parabola. In this lesson, you will graph Quadratic Functions , where b and c equal 0. Previous domain range Core Concept vertical shrink Characteristics of Quadratic Functions vertical stretch The parent Quadratic function is f (x) = x2. The graphs of all other Quadratic reflection Functions are transformations of the graph of the parent Quadratic function. The lowest point y The vertical line that on a parabola that divides the parabola opens up or the into two symmetric highest point on a parts is the axis of REMEMBER decreasing increasing parabola that opens symmetry.

10 The axis The notation f (x) is down is the vertex. of symmetry passes another name for y. The vertex of the axis of x through the vertex. For graph of f (x) = x2 vertex the graph of f (x) = x2, symmetry is (0, 0). the axis of symmetry is the y-axis, or x = 0. Identifying Characteristics of a Quadratic Function y Consider the graph of the Quadratic function. 6. Using the graph, you can identify characteristics such as the vertex, axis of symmetry, 4 and the behavior of the graph, as shown. You can also determine the following: 2.