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**Grade 8 Algebra I Identifying Quadratic Functions**

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**Factor each polynomial completely. Check your answer.**

Warm Up Factor each polynomial completely. Check your answer. 1) 3x2 + 7x + 4 2) 2p5 + 10p4 - 12p3 3) 9q6 + 30q5 + 24q4 CONFIDENTIAL

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**The function y = x2 can be written as y = 1x2 + 0x + 0,**

Quadratic Functions A quadratic function is any function that can be written in the standard form y = ax2 + bx + c, where a, b, and c are real numbers and a ≠ 0. -3 3 y = x2 x y The function y =x2 is shown in the graph. Notice that the graph is not linear. This function is a quadratic function. The function y = x2 can be written as y = 1x2 + 0x + 0, where a = 1, b = 0, and c = 0. CONFIDENTIAL

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**Constant change in x-values**

You can see that a constant change in x corresponded to a constant change in y. The differences between y-values for a constant change in x-values are called first differences. + 1 Constant change in x-values + 3 + 5 + 7 + 2 First differences Second differences x 1 2 3 4 y = x2 9 16 Notice that the quadratic function y = x2 does not have constant first differences. It has constant second differences. This is true for all quadratic functions. CONFIDENTIAL

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**Identifying Quadratic Functions**

Tell whether each function is quadratic. Explain. A) Since you are given a table of ordered pairs with a constant change in x-values, see if the second differences are constant. Find the first differences, then find the second differences. + 2 -6 -2 +2 +6 +4 x -4 -2 2 4 y 8 The function is quadratic. The second differences are constant. CONFIDENTIAL

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**Since you are given an equation, use y = ax2 + bx + c.**

B) y = -3x + 20 Since you are given an equation, use y = ax2 + bx + c. This is not a quadratic function because the value of a is 0. C) y + 3x2 = -4 y + 3x2 = -4 3x x2 Try to write the function in the form y = ax2 + bx + c by solving for y. Subtract 3x2 from both sides. y = -3x2 - 4 This is a quadratic function because it can be written in the form y = ax2 + bx + c where a = -3, b = 0, and c = -4. CONFIDENTIAL

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**Tell whether each function is quadratic. Explain.**

Now you try! Tell whether each function is quadratic. Explain. 1) (-2, 4) , (-1, 1) , (0, 0) , (1, 1) , (2, 4) 2) y + x = 2x2 CONFIDENTIAL

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**The graph of a quadratic function is a curve called a parabola . **

To graph a quadratic function, generate enough ordered pairs to see the shape of the parabola. Then connect the points with a smooth curve. x 3 -3 y CONFIDENTIAL

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**Graphing Quadratic Functions by Using a Table of Values**

Use a table of values to graph each quadratic function. 1) y = 2x2 Make a table of values. Choose values of x and use them to find values of y. x -2 -1 1 2 y = 2x2 8 Graph the points. Then connect the points with a smooth curve. 0,0 2 4 -2,8 2,8 -1,2 1,2 6 8 x y CONFIDENTIAL

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**Graph the points. Then connect the points with a smooth curve.**

2) y = -2x2 Make a table of values. Choose values of x and use them to find values of y. x -2 -1 1 2 y = -2x2 -8 Graph the points. Then connect the points with a smooth curve. 0,0 -2 4 2,-8 -2,-8 -1,2 -1,-2 -4 -6 -8 x y CONFIDENTIAL

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**When a quadratic function is written in the form y = ax2 + bx + c, **

As shown in the graphs in Examples 1 and 2, some parabolas open upward and some open downward. Notice that the only difference between the two equations is the value of a. When a quadratic function is written in the form y = ax2 + 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. CONFIDENTIAL

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**Identifying the Direction of a Parabola**

Tell whether the graph of each quadratic function opens upward or downward. Explain. A) y = 4x2 y = 4x2 a = 4 Identify the value of a. Since a > 0, the parabola opens upward. B) 2x2+ y = 5 2x2+ y = 5 Write the function in the form y = ax2 + bx + c by solving for y. Subtract 2x2 from both sides. 2x x2 y = -2x2 + 5 a = -2 Identify the value of a. Since a < 0, the parabola opens downward. CONFIDENTIAL

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Now you try! Tell whether the graph of each quadratic function opens upward or downward. Explain. 1) f (x) = -4x2 - x + 1 2) y - 5x2 = 2x - 6 CONFIDENTIAL

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**Minimum and Maximum Values**

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. WORDS If a > 0, the parabola opens upward, and the y-value of the vertex is the minimum value of the function. If a < 0, the parabola opens downward, and the y-value of the vertex is the maximum value of the function. GRAPHS y = x2 + 6x + 9 y = x2 + 6x - 4 y y 4 vertex: (3,5) maximum: 5 4 vertex: (-3,0) minimum: 0 2 2 x x -4 -2 2 4 CONFIDENTIAL

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**Identifying the Vertex and the Minimum or Maximum**

Identify the vertex of each parabola. Then give the minimum or maximum value of the function. -4 -2 2 x y 2 4 -2 x y The vertex is (1, 5) ,and the maximum is 5. The vertex is (-2, -5) , and the minimum is -5. CONFIDENTIAL

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Now you try! Identify the vertex of the parabola. Then give the minimum or maximum value of the function. y 4 2 x -2 CONFIDENTIAL

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**the range begins at the minimum value of the function, where y = 1. **

Unless a specific domain is given, you may assume that the domain of a quadratic function is all real numbers. You can find the range of a quadratic function by looking at its graph. 2 4 x y 6 For the graph of y = x2 - 4x + 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. CONFIDENTIAL

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**Finding Domain and Range**

Find the domain and range. -2 2 x y 4 Step1: The graph opens downward, so identify the maximum. The vertex is (-1, 4) , so the maximum is 4. Step2: Find the domain and range. D: all real numbers R: y ≤ 4 CONFIDENTIAL

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**Find the domain and range.**

Now you try! Find the domain and range. y 4 2 x -2 -4 -6 CONFIDENTIAL

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**Tell whether each function is quadratic. Explain:**

Assessment Tell whether each function is quadratic. Explain: 1) y + 6x = -14 2) 2x2 + y = 3x - 1 3) x -4 -3 -2 -1 y 39 18 3 -6 -9 4) (-10, 15) , (-9, 17) , (-8, 19) , (-7, 21) , (-6, 23) CONFIDENTIAL

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**Use a table of values to graph each quadratic function.**

5) y = 4x2 6) y = 1x2 2 CONFIDENTIAL

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**Tell whether the graph of each quadratic function opens upward or downward. Explain.**

7) y = -3x2 + 4x 8) y = 1 - 2x + 6x2 CONFIDENTIAL

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**9) Identify the vertex of the parabola**

9) Identify the vertex of the parabola. Then give the minimum or maximum value of the function. 4 2 x y -2 CONFIDENTIAL

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**10) Find the domain and range.**

4 2 x y -2 CONFIDENTIAL

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**The function y = x2 can be written as y = 1x2 + 0x + 0,**

Let’s review Quadratic Functions A quadratic function is any function that can be written in the standard form y = ax2 + bx + c, where a, b, and c are real numbers and a ≠ 0. -3 3 y = x2 x y The function y =x2 is shown in the graph. Notice that the graph is not linear. This function is a quadratic function. The function y = x2 can be written as y = 1x2 + 0x + 0, where a = 1, b = 0, and c = 0. CONFIDENTIAL

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**Constant change in x-values**

You can see that a constant change in x corresponded to a constant change in y. The differences between y-values for a constant change in x-values are called first differences. + 1 Constant change in x-values + 3 + 5 + 7 + 2 First differences Second differences x 1 2 3 4 y = x2 9 16 Notice that the quadratic function y = x2 does not have constant first differences. It has constant second differences. This is true for all quadratic functions. CONFIDENTIAL

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**Graphing Quadratic Functions by Using a Table of Values**

Use a table of values to graph each quadratic function. 1) y = 2x2 Make a table of values. Choose values of x and use them to find values of y. x -2 -1 1 2 y = 2x2 8 Graph the points. Then connect the points with a smooth curve. 0,0 2 4 -2,8 2,8 -1,2 1,2 6 8 x y CONFIDENTIAL

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**Identifying the Direction of a Parabola**

Tell whether the graph of each quadratic function opens upward or downward. Explain. A) y = 4x2 y = 4x2 a = 4 Identify the value of a. Since a > 0, the parabola opens upward. B) 2x2+ y = 5 2x2+ y = 5 Write the function in the form y = ax2 + bx + c by solving for y. Subtract 2x2 from both sides. 2x x2 y = -2x2 + 5 a = -2 Identify the value of a. Since a < 0, the parabola opens downward. CONFIDENTIAL

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**Minimum and Maximum Values**

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. WORDS If a > 0, the parabola opens upward, and the y-value of the vertex is the minimum value of the function. If a < 0, the parabola opens downward, and the y-value of the vertex is the maximum value of the function. GRAPHS y = x2 + 6x + 9 y = x2 + 6x - 4 y y 4 vertex: (3,5) maximum: 5 4 vertex: (-3,0) minimum: 0 2 2 x x -4 -2 2 4 CONFIDENTIAL

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**Finding Domain and Range**

Find the domain and range. -2 2 x y 4 Step1: The graph opens downward, so identify the maximum. The vertex is (-1, 4) , so the maximum is 4. Step2: Find the domain and range. D: all real numbers R: y ≤ 4 CONFIDENTIAL

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**You did a great job today!**

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