9 1 Identifying Quadratic Functions Warm Up 1

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9 -1 Identifying Quadratic Functions Warm Up 1. Evaluate x 2 + 5 x

9 -1 Identifying Quadratic Functions Warm Up 1. Evaluate x 2 + 5 x for x = 4 and x = – 3. 36; – 6 2. Generate ordered pairs for the function y = x 2 + 2 with the given domain. D: {– 2, – 1, 0, 1, 2} Holt Algebra 1 x – 2 – 1 0 1 2 y 6 3 2 3 6

9 -1 Identifying Quadratic Functions Holt Algebra 11

9 -1 Identifying Quadratic Functions Holt Algebra 11

9 -1 Identifying Quadratic Functions The function y = x 2 is shown in

9 -1 Identifying Quadratic Functions The function y = x 2 is shown in the graph. Notice that the graph is not linear. A quadratic function is any function that can be written in the standard form y = ax 2 + bx + c, where a, b, and c are real numbers and a ≠ 0. The function y = x 2 can be written as y = 1 x 2 + 0 x + 0, where a = 1, b = 0, and c = 0. Holt Algebra 1

9 -1 Identifying Quadratic Functions Notice that the quadratic function y = x 2

9 -1 Identifying Quadratic Functions Notice that the quadratic function y = x 2 has constant second differences. Holt Algebra 1

9 -1 Identifying Quadratic Functions Example 1 A: Identifying Quadratic Functions Tell whether the

9 -1 Identifying Quadratic Functions Example 1 A: Identifying Quadratic Functions Tell whether the function is quadratic. Explain. Since you are given a table x y of ordered pairs with a constant change in x– 2 – 9 +7 +1 values, see if the second – 6 – 1 – 2 differences are constant. +1 +1 +0 0 – 1 +1 +1 1 0 2 7 +1 +7 +6 Find the first differences, then find the second differences. The function is not quadratic. The second differences are not constant. Holt Algebra 1

9 -1 Identifying Quadratic Functions Example 1 B: Identifying Quadratic Functions Tell whether the

9 -1 Identifying Quadratic Functions Example 1 B: Identifying Quadratic Functions Tell whether the function is quadratic. Explain. y = 7 x + 3 Since you are given an equation, use y = ax 2 + bx + c. This is not a quadratic function because the value of a is 0. Holt Algebra 1

9 -1 Identifying Quadratic Functions Example 1 C: Identifying Quadratic Functions Tell whether the

9 -1 Identifying Quadratic Functions Example 1 C: Identifying Quadratic Functions Tell whether the function is quadratic. Explain. y – 10 x 2 = 9 10 x 2 y– =9 + 10 x 2 +10 x 2 y = 10 x 2 + 9 Try to write the function in the form y = ax 2 + bx + c by solving for y. Add 10 x 2 to both sides. This is a quadratic function because it can be written in the form y = ax 2 + bx + c where a = 10, b = 0, and c =9. Holt Algebra 1

9 -1 Identifying Quadratic Functions The graph of a quadratic function is a curve

9 -1 Identifying Quadratic Functions The graph of a quadratic function is a curve called a parabola. Holt Algebra 1

9 -1 Identifying Quadratic Functions Example 2 B: Graphing Quadratic Functions by Using a

9 -1 Identifying Quadratic Functions Example 2 B: Graphing Quadratic Functions by Using a Table of Values Use a table of values to graph the quadratic function. y = – 4 x 2 x y – 2 – 16 – 1 – 4 0 0 1 – 4 2 – 16 Holt Algebra 1 Make a table of values. Choose values of x and use them to find values of y. Graph the points. Then connect the points with a smooth curve.

9 -1 Identifying Quadratic Functions Check It Out! Example 2 a Use a table

9 -1 Identifying Quadratic Functions Check It Out! Example 2 a Use a table of values to graph each quadratic function. y = x 2 + 2 x y – 2 6 – 1 3 0 2 1 3 2 6 Holt Algebra 1 Make a table of values. Choose values of x and use them to find values of y. Graph the points. Then connect the points with a smooth curve.

9 -1 Identifying Quadratic Functions When a quadratic function is written in the form

9 -1 Identifying Quadratic Functions When a quadratic function is written in the form y = ax 2 + bx + c, the value of a determines the direction a parabola opens. • A parabola opens upward when a > 0. • A parabola opens downward when a < 0. Holt Algebra 1

9 -1 Identifying Quadratic Functions Example 3 A: Identifying the Direction of a Parabola

9 -1 Identifying Quadratic Functions Example 3 A: Identifying the Direction of a Parabola Tell whether the graph of the quadratic function opens upward or downward. Explain. Write the function in the form y = ax 2 + bx + c by solving for y. Add to both sides. Identify the value of a. Since a > 0, the parabola opens upward. Holt Algebra 1

9 -1 Identifying Quadratic Functions Example 3 B: Identifying the Direction of a Parabola

9 -1 Identifying Quadratic Functions Example 3 B: Identifying the Direction of a Parabola Tell whether the graph of the quadratic function opens upward or downward. Explain. y = 5 x – 3 x 2 y = – 3 x 2 + 5 x Write the function in the form y = ax 2 + bx + c. a = – 3 Identify the value of a. Since a < 0, the parabola opens downward. Holt Algebra 1

9 -1 Identifying Quadratic Functions The highest or lowest point on a parabola is

9 -1 Identifying Quadratic Functions The highest or lowest point on a parabola is the vertex. If a parabola opens upward, the vertex is the lowest point. If a parabola opens downward, the vertex is the highest point. Holt Algebra 1

9 -1 Identifying Quadratic Functions Holt Algebra 1

9 -1 Identifying Quadratic Functions Holt Algebra 1

9 -1 Identifying Quadratic Functions Identify the vertex of each parabola. Then give the

9 -1 Identifying Quadratic Functions Identify the vertex of each parabola. Then give the minimum or maximum value of the function. A. B. The vertex is (– 3, 2), and the minimum is 2. Holt Algebra 1 The vertex is (2, 5), and the maximum is 5.

9 -1 Identifying Quadratic Functions Check It Out! Example 4 Identify the vertex of

9 -1 Identifying Quadratic Functions Check It Out! Example 4 Identify the vertex of each parabola. Then give the minimum or maximum value of the function. a. b. The vertex is (– 2, 5) and the maximum is 5. Holt Algebra 1 The vertex is (3, – 1), and the minimum is – 1.

9 -1 Identifying Quadratic Functions For the graph of y = x 2 –

9 -1 Identifying Quadratic Functions For the graph of y = x 2 – 4 x + 5, the range begins at the minimum value of the function, where y = 1. All the y-values of the function are greater than or equal to 1. So the range is y 1. Holt Algebra 1

9 -1 Identifying Quadratic Functions Example 5: Finding Domain and Range Find the domain

9 -1 Identifying Quadratic Functions Example 5: Finding Domain and Range Find the domain and range. Step 1 The graph opens downward, so identify the maximum. The vertex is (– 5, – 3), so the maximum is – 3. Step 2 Find the domain and range. D: all real numbers R: y ≤ – 3 Holt Algebra 1

9 -1 Identifying Quadratic Functions 3. Holt Algebra 1

9 -1 Identifying Quadratic Functions 3. Holt Algebra 1

9 -1 Identifying Quadratic Functions Lesson Quiz: Part I 1. Is y = –x

9 -1 Identifying Quadratic Functions Lesson Quiz: Part I 1. Is y = –x – 1 quadratic? Explain. No; there is no x 2 -term, 2. Graph y = 1. 5 x 2. so a = 0. Holt Algebra 1

9 -1 Identifying Quadratic Functions Lesson Quiz: Part II Use the graph for Problems

9 -1 Identifying Quadratic Functions Lesson Quiz: Part II Use the graph for Problems 3 -5. 3. Identify the vertex. (5, – 4) 4. Does the function have a minimum or maximum? What is it? max; – 4 5. Find the domain and range. D: all real numbers; R: y ≤ – 4 Holt Algebra 1