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**Graphs of Quadratic Functions**

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Quadratic Functions: Definition of Quadratic Function: The graph of a quadratic function is a special type of “U”-shaped curve that is called a parabola. (U shaped) All parabolas are symmetric with respect to a line called the axis of symmetry, or simply the axis of the parabola. The point where the axis intersects the parabola is the vertex of the parabola.

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**Quadratic Equations y = ax2 + bx + c y = a(x - h)2 + k f(x) =**

Standard form Vertex form (later) The graph is “U-shaped” and is called a parabola.

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**The highest or lowest point on the parabola is called the vertex.**

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**The axis of symmetry for the parabola is the vertical line through the vertex.**

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**y = x2 y = -x2 Vertex is a minimum Vertex is a maximum Opens up**

Opens down

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**Finding the vertex of a parabola:**

This gives you the x-value and plug the x-value into the original function to find the y-value

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EXAMPLES: Sketch the graph of the quadratic function without using a graphing utility. Identify the vertex and x-intercepts. You know the graph will go up b/c a is a positive number Find the vertex first!! So the x part of the vertex is -2 Now, to find the y-value, plug in -2 wherever you see an x. Vertex: (-2,-7)

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**Now make the table! Remember the vertex goes in the middle!!! (x, y) x**

-2

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**Sketch the graph of the quadratic function without using a graphing **

EXAMPLES: Sketch the graph of the quadratic function without using a graphing utility. Identify the vertex and x-intercepts. You know the graph will go up b/c a is a positive number Find the vertex first!! Where is B??? So the x part of the vertex is 0 Now, to find the y-value, plug in zero wherever you see an x. Vertex: (0,-5)

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**So, we know that the vertex is (0,-5) now all we have to do is a**

T-chart and pick 4 more points and then graph. x y 0 -5

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One More…. Name the vertex, axis of symmetry and whether it opens up or down AND GRAPH. y = 2x2 + 1 Vertex: (0, 1) a = 2 b = 0 c = 1 Axis of sym: x = 0 Opens up

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**Name the vertex, axis of symmetry and whether it opens up or down**

AND GRAPH. y = 2x2 + 1 Vertex: (0, 1) x y Put vertex in the middle of the t-table -2 9 -1 3 1 1 3 2 9

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**Example Graph the quadratic function f (x) = -x2 + 6x -. Solution:**

Step Determine how the parabola opens. Note that a, the coefficient of x 2, is -1. Thus, a < 0; this negative value tells us that the parabola opens downward. Step Find the vertex. We know the x-coordinate of the vertex is –b/2a. We identify a, b, and c to substitute the values into the equation for the x-coordinate: x = -b/(2a) = -6/2(-1) = 3. The x-coordinate of the vertex is 3. We substitute 3 for x in the equation of the function to find the y-coordinate: the parabola has its vertex at (3,7).

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**Example Graph the quadratic function f (x) = -x2 + 6x -.**

Step Find the x-intercepts. Replace f (x) with 0 in f (x) = -x2 + 6x = -x2 + 6x - 2

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**Example Graph the quadratic function f (x) = -x2 + 6x -.**

Step Find the y-intercept. Replace x with 0 in f (x) = -x2 + 6x - 2. f (0) = • = - The y-intercept is –2. The parabola passes through (0, -2). Step Graph the parabola.

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**Minimum and Maximum: Quadratic Functions**

Consider f(x) = ax2 + bx +c. If a > 0, then f has a minimum that occurs at x = -b/(2a). This minimum value is f(-b/(2a)). If a < 0, the f has a maximum that occurs at x = -b/(2a). This maximum value is f(-b/(2a)).

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**Strategy for Solving Problems Involving Maximizing or Minimizing Quadratic Functions**

Read the problem carefully and decide which quantity is to be maximized or minimized. Use the conditions of the problem to express the quantity as a function in one variable. Rewrite the function in the form f(x) = ax2 + bx +c. Calculate -b/(2a). If a > 0, then f has a minimum that occurs at x = -b/(2a). This minimum value is f(-b/(2a)). If a < 0, the f has a maximum that occurs at x = -b/(2a). This maximum value is f(-b/(2a)). Answer the question posed in the problem.

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