Lesson 8-1 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 is not 0.

The differences between y-values for a constant change in x-values are called first differences.

Quadratic functions have constant second differences. What is a quadratic

function? 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. What is a parabola? When a quadratic function is written in standard form,

y = ax^2 + bx + c, the value of a determines the direction a parabola opens.

When a > 0, the parabola opens upward.

When a < 0, the parabola opens downward. How do we know which direction the parabola opens? The highest or lowest point on a parabola is the vertex.

If a parabola opens upward (a > 0), the vertex is the lowest point.

The y-value of the vertex is the minimum value of the function

If a parabola opens downward (a < 0), the vertex is the highest point.

The y-value of the vertex is the maximum value of the function. What is the vertex of a parabola? Unless a specific domain is given, you can 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.

If the graph has a maximum, the range is less than or equal to that value. If the graph has a minimum, the range is greater than or equal to that value. What is the domain and range of a quadratic function? p526 - 529

(2 - 38 even,

54 - 60 even,

65 - 67) Assignment Identifying Quadratic

Functions –6

+0

+6 +1

+1

+1

+1 -7

-1

-1

+7 7 –9 –2 2 -–1 -–2 y x 1 0 0 –1 The function is not quadratic. The second differences are not constant. Find the first differences, then find the second differences. Since you are given a table of ordered pairs with a constant change in x-values, see if the second differences are constant. Tell whether the function is quadratic. Explain. Example 1A: Identifying Quadratic Functions This is not a quadratic function because the value of a is 0. y = 7x + 3 Tell whether the function is quadratic. Explain. Example 1B: Identifying Quadratic Functions Since you are given an equation, use y = ax^2 + bx + c. +2

+2

+2 +1

+1

+1

+1 –3

–1

+1

+3 4 4 1 1 0 2 1 0 y x -–1 –-2 The function is quadratic. The second differences are constant. Find the first differences, then find the second differences. List the ordered pairs in a table of values. Since there is a constant change in the x-values, see if the differences are constant. Tell whether the function is quadratic. Explain.

{(–-2, 4), (-–1, 1), (0, 0), (1, 1), (2, 4)} Check It Out! Example 1a Check It Out! Example 1b y + x = 2x^2 y = 2x^2 – x – x – x Try to write the function in the form y = ax2 + bx + c by solving for y. Subtract x from both sides. y + x = 2x^2 This is a quadratic function because it can be written

in the form y = ax^2 + bx + c where a = 2, b = –1, and c = 0. Tell whether the function is quadratic. Explain. Example 2: Use a table of values to graph a quadratic function. a) y = x^2 + 2 b) y = -3x^2 + 1

x -2 -1 0 1 2 x -2 -1 0 1 2

y __ __ __ __ __ y __ __ __ __ __ Graph these two quadratics on your paper. Example 3: Tell whether the graph of each quadratic function opens upward or downward. Explain. a) y = 4x^2 b) 2x^2 + y = 5

c) f(x) = -4x^2 - x + 1 d) y - 5x^2 = 2x - 6 Example 4: Identify the vertex. Then give the minimum or maximum value of the function. a) Identify the vertex of each parabola. Then give the minimum or maximum value of the function. B) C) D) Example 5: Find the Domain and Range of Each Quadratic Function. a) B) C)

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# Lesson 8-1 Identify Quadratic Functions

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