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**THE GRAPH OF A QUADRATIC FUNCTION**

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**QUADRATIC FUNCTIONS A quadratic function is a function of the form**

f(x) = ax 2 + bx + c where a, b & c are real numbers and a 0 The domain of a quadratic function is all real numbers.

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**GRAPHS OF QUADRATIC FUNCTIONS**

As we’ve already seen, f(x) = x2 graphs into a PARABOLA. This is the simplest quadratic function we can think of. We will use this one as a model by which to compare all other quadratic functions we will examine.

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VERTEX OF A PARABOLA All parabolas have a VERTEX, the lowest or highest point on the graph (depending upon whether it opens up or down.)

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AXIS OF SYMMETRY All parabolas have an AXIS OF SYMMETRY, an imaginary line which goes through the vertex and about which the parabola is symmetric.

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**HOW PARABOLAS DIFFER Some parabolas open up and some open down.**

Parabolas will all have a different vertex and a different axis of symmetry. Some parabolas will be wide and some will be narrow.

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**y = (x - 3)2 - 4 Vertex x y 6 5 5 4 -3 3 -4 2 -3 1 5 Example**

Y-intercept x y Axis of Symmetry 6 5 5 Roots or x intercepts 4 -3 3 -4 Vertex 2 -3 1 5

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**GRAPHS OF QUADRATIC FUNCTIONS**

The general form of a quadratic function is: f(x) = ax2 + bx + c The position, width, and orientation of a particular parabola will depend upon the values of a, b, and c.

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**GRAPHS OF QUADRATIC FUNCTIONS**

Compare f(x) = x2 to the following: f(x) = 2x2 f(x) = .5x f(x) = -.5x2 If a > 0, then the parabola opens up If a < 0, then the parabola opens down

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**GRAPHS OF QUADRATIC FUNCTIONS**

Now compare f(x) = x2 to the following: f(x) = x f(x) = x 2 - 2 Vertical shift up Vertical shift down

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**GRAPHS OF QUADRATIC FUNCTIONS**

Now compare f(x) = x2 to the following: f(x) = (x + 2)2 f(x) = (x – 3)2 Horizontal shift to the left Horizontal shift to the right

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**GRAPHS OF QUADRATIC FUNCTIONS**

When the general form of a quadratic function f(x) = ax2 + bx + c is changed to the vertex form: f(x) = a(x - h) 2 + k We can tell by horizontal and vertical shifting of the parabola where the vertex will be. The parabola will be shifted h units horizontally and k units vertically.

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**GRAPHS OF QUADRATIC FUNCTIONS**

Thus, a quadratic function written in the form f(x) = a(x - h) 2 + k will have a vertex at the point (h,k). The value of “a” will determine whether the parabola opens up or down (positive or negative) and whether the parabola is narrow or wide.

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**GRAPHS OF QUADRATIC FUNCTIONS**

f(x) = a(x - h) 2 + k Vertex (highest or lowest point): (h,k) If a > 0, then the parabola opens up If a < 0, then the parabola opens down

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**GRAPHS OF QUADRATIC FUNCTIONS**

Axis of Symmetry The vertical line about which the graph of a quadratic function is symmetric. x = h where h is the x-coordinate of the vertex.

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**GRAPHS OF QUADRATIC FUNCTIONS**

So, if we want to examine the characteristics of the graph of a quadratic function, our job is to transform the general form: f(x) = ax2 + bx + c into the vertex form: f(x) = a(x – h)2 + k

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**GRAPHS OF QUADRATIC FUNCTIONS**

This will require to process of completing the square which is a little different than completing the square to solve a quadratic equation.

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**Graphing Quadratic Functions by Completing the Square**

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**Remember about Perfect Square Trinomials**

Factor x2 + 6x + 9 (x + 3)(x + 3) or (x + 3)2 Perfect Square Trinomial The factors are in the form (x + a)2 or (x - a)2. Note the relationship between the middle term and the last term. The last term is one-half the middle term squared. = 32 = 9 Find the value of the last term that will make the following perfect square trinomials. (x + 7)2 x2 + 14x + _______ 49 x2 + 7x + _______ x2 - 3x + _______

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**Changing from Standard Form to Vertex Form**

Write y = x2 + 10x + 23 in the form y = a(x - h)2 + k. Sketch the graph. y = x x ( ) 1. Bracket the first two terms. y = (x2 + 10x + ____ - ____) + 23 25 25 2. Add a value within the brackets to make a perfect square trinomial. Whatever you add must be subtracted to keep the value of the function the same. y = (x2 + 10x + 25) y = (x + 5)2 - 2 3. Group the perfect square trinomial. (-5, -2) 4. Factor the trinomial and simplify.

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**Changing from Standard Form to Vertex Form**

Write y = 2x2 - 12x -11 in the form y = a(x - h)2 + k. Sketch the graph. y = 2x x ( ) 1. Bracket the first two terms. y = 2(x2 - 6x) 2. Factor out the coefficient of the x2- term. y = 2(x2 - 6x + ____ - ____) - 11 9 9 y = 2(x2 - 6x + 9) 3. Add a value within the brackets to make a perfect square trinomial. Whatever you add must be subtracted to keep the value of the function the same. y = 2(x - 3)2 - 29 Multiply, when you remove this term from the brackets. 4. Group the perfect square trinomial. When grouping the trinomial, remember to distribute the coefficient. (3, -29) 5. Factor the trinomial and simplify.

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Completing the Square y = -3x2 + 5x - 1 y = -3x2 + 5x - 1 ( ) y = -3(x x) - 1 y = -3(x x + ______ - ______ ) - 1 y = -3(x x ) y = -3(x ) y = -3(x )2 + Vertex is y = -3(x )2 +

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The Vertex Formula

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**Using the general form, y = ax2 + bx + c, complete the square:**

Completing the Square - The General Case Using the general form, y = ax2 + bx + c, complete the square: y = ax2 + bx + c y = (ax2 + bx ) + c The vertex is This IS the vertex BUT it is easier just to remember that the x-value is and then plug that in to the equation to get the y-value for the vertex.

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**Using the Vertex Formula**

Find the vertex and the maximum or minimum value of f(x) = -4x2 - 12x + 5 using the axis of symmetry, the vertex is Find the x-value of the vertex: Find the y-value of the vertex: Therefore there is a maximum of y = 14, when x = The vertex is

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**Direction of the Parabola**

If the coefficient of x2 is positive the parabola will open up. If the coefficient of x2 is negative the parabola will open down.

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**CHARACTERISTICS OF THE GRAPH OF A QUADRATIC FUNCTION**

f(x) = ax2 + bx + c Parabola opens up and has a minimum value if a > 0. Parabola opens down and has a maximum value if a < 0.

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EXAMPLE Determine without graphing whether the given quadratic function has a maximum or minimum value and then find the value. Verify by graphing. f(x) = 4x2 - 8x + 3 g(x) = -2x2 + 8x + 3

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**THE X AND Y INTERCEPTS OF A QUADRATIC FUNCTION**

Find the x-intercepts by setting the quadratic function equal to zero and solve by whatever method is easiest. If the discriminant b2 – 4ac > 0, the graph of f(x) = ax2+ bx + c has two distinct x-intercepts and will cross the x-axis twice. 3. If the discriminant b2 – 4ac = 0, the graph of f(x) = ax2 + bx + c has one x-intercept and touches the x-axis at its vertex. 4. If the discriminant b2 – 4ac < 0, the graph of f(x) = ax2 + bx + c has no x-intercept and will not cross or touch the x-axis. 5. Find the y-intercept by substituting x=0 into function.

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**GRAPHING QUADRATIC FUNCTIONS**

Graph the functions below by hand by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. Verify your results using a graphing calculator. f(x) = 2x g(x) = x2 - 6x - 1 h(x) = 3x2 + 6x k(x) = -2x2 + 6x + 2

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