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CONFIDENTIAL 1 Grade 8 Algebra I Identifying Quadratic Functions

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CONFIDENTIAL 2 Warm Up Factor each polynomial completely. Check your answer. 1) 3x 2 + 7x + 4 2) 2p 5 + 10p 4 - 12p 3 3) 9q 6 + 30q 5 + 24q 4

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CONFIDENTIAL 3 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. Quadratic Functions -3 3 3 0 y = x 2 x y The function y =x 2 is shown in the graph. Notice that the graph is not linear. This function is a quadratic function. The function y = x 2 can be written as y = 1x 2 + 0x + 0, where a = 1, b = 0, and c = 0.

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CONFIDENTIAL 4 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. x01234 y = x 2 014916 + 1 Constant change in x-values + 1 + 3 + 5 + 7 + 2 First differences Second differences Notice that the quadratic function y = x 2 does not have constant first differences. It has constant second differences. This is true for all quadratic functions.

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CONFIDENTIAL 5 Identifying Quadratic Functions Tell whether each function is quadratic. Explain. Since you are given a table of ordered pairs with a constant change in x-values, see if the second differences are constant. x-4-2024 y82028 + 2 -6 -2 +2 +6 +4 Find the first differences, then find the second differences. The function is quadratic. The second differences are constant. A)

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CONFIDENTIAL 6 B) y = -3x + 20 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. C) y + 3x 2 = -4 This is a quadratic function because it can be written in the form y = ax 2 + bx + c where a = -3, b = 0, and c = -4. y + 3x 2 = -4 3x 2 3x 2 y = -3x 2 - 4 Try to write the function in the form y = ax 2 + bx + c by solving for y. Subtract 3x 2 from both sides.

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CONFIDENTIAL 7 Tell whether each function is quadratic. Explain. Now you try! 1) (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4) 2) y + x = 2x 2

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CONFIDENTIAL 8 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. 0 x 3 -3 y

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CONFIDENTIAL 9 Graphing Quadratic Functions by Using a Table of Values Use a table of values to graph each quadratic function. 1) y = 2x 2 x-2012 y = 2x 2 82028 0,0244 -2,82,8 -1,2 1,2 2 4 6 8 x y 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.

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CONFIDENTIAL 10 2) y = -2x 2 x-2012 y = -2x 2 -8-20 -8 0,0-244 2,-8-2,-8 -1,2 -1,-2 -2 -4 -6 -8 x y 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.

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CONFIDENTIAL 11 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 = 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.

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CONFIDENTIAL 12 Tell whether the graph of each quadratic function opens upward or downward. Explain. Identifying the Direction of a Parabola A) y = 4x 2 y = 4x 2 a = 4 Identify the value of a. Since a > 0, the parabola opens upward. B) 2x 2 + y = 5 2x 2 + y = 5 a = -2 Write the function in the form y = ax 2 + bx + c by solving for y. Subtract 2x 2 from both sides. Since a < 0, the parabola opens downward. Identify the value of a. 2x 2 y = -2x 2 + 5

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CONFIDENTIAL 13 Now you try! 1) f (x) = -4x 2 - x + 1 2) y - 5x 2 = 2x - 6 Tell whether the graph of each quadratic function opens upward or downward. Explain.

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CONFIDENTIAL 14 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 = x 2 + 6x + 9y = x 2 + 6x - 4 2 4 -2 2 4 00 -4 vertex: (-3,0) minimum: 0 24 x x y y vertex: (3,5) maximum: 5

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CONFIDENTIAL 15 Identifying the Vertex and the Minimum or Maximum Identify the vertex of each parabola. Then give the minimum or maximum value of the function. 2 4 2 0 -2 x y -4 -2 02 x y The vertex is (1, 5),and the maximum is 5. The vertex is (-2, -5), and the minimum is -5.

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CONFIDENTIAL 16 Identify the vertex of the parabola. Then give the minimum or maximum value of the function. Now you try! 2 4 0 -2 x y

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CONFIDENTIAL 17 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. For the graph of y = x 2 - 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. 2 4 402 x y 6

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CONFIDENTIAL 18 Finding Domain and Range Find the domain and range. -2 2 2 0 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

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CONFIDENTIAL 19 Find the domain and range. Now you try! -2 -4 4 0 2 x y -6

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CONFIDENTIAL 20 BREAK

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CONFIDENTIAL 21

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CONFIDENTIAL 22 Assessment 1) y + 6x = -14 2) 2x 2 + y = 3x - 1 Tell whether each function is quadratic. Explain: x-4-3-20 y39183-6-9 3) 4) (-10, 15), (-9, 17), (-8, 19), (-7, 21), (-6, 23)

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CONFIDENTIAL 23 5) y = 4x 2 6) y = 1x 2 2 Use a table of values to graph each quadratic function.

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CONFIDENTIAL 24 7) y = -3x 2 + 4x Tell whether the graph of each quadratic function opens upward or downward. Explain. 8) y = 1 - 2x + 6x 2

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CONFIDENTIAL 25 9) Identify the vertex of the parabola. Then give the minimum or maximum value of the function. 4 0 2 x y 2 -2

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CONFIDENTIAL 26 10) Find the domain and range. 4 0 2 x y 2 -2

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CONFIDENTIAL 27 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. Quadratic Functions -3 3 3 0 y = x 2 x y The function y =x 2 is shown in the graph. Notice that the graph is not linear. This function is a quadratic function. The function y = x 2 can be written as y = 1x 2 + 0x + 0, where a = 1, b = 0, and c = 0. Let’s review

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CONFIDENTIAL 28 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. x01234 y = x 2 014916 + 1 Constant change in x-values + 1 + 3 + 5 + 7 + 2 First differences Second differences Notice that the quadratic function y = x 2 does not have constant first differences. It has constant second differences. This is true for all quadratic functions.

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CONFIDENTIAL 29 Graphing Quadratic Functions by Using a Table of Values Use a table of values to graph each quadratic function. 1) y = 2x 2 x-2012 y = 2x 2 82028 0,0244 -2,82,8 -1,2 1,2 2 4 6 8 x y 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.

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CONFIDENTIAL 30 Tell whether the graph of each quadratic function opens upward or downward. Explain. Identifying the Direction of a Parabola A) y = 4x 2 y = 4x 2 a = 4 Identify the value of a. Since a > 0, the parabola opens upward. B) 2x 2 + y = 5 2x 2 + y = 5 a = -2 Write the function in the form y = ax 2 + bx + c by solving for y. Subtract 2x 2 from both sides. Since a < 0, the parabola opens downward. Identify the value of a. 2x 2 y = -2x 2 + 5

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CONFIDENTIAL 31 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 = x 2 + 6x + 9y = x 2 + 6x - 4 2 4 -2 2 4 00 -4 vertex: (-3,0) minimum: 0 24 x x y y vertex: (3,5) maximum: 5

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CONFIDENTIAL 32 Finding Domain and Range Find the domain and range. -2 2 2 0 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

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CONFIDENTIAL 33 You did a great job today!

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