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SECTION 2.4 THE GRAPH OF A QUADRATIC FUNCTION THE GRAPH OF A QUADRATIC FUNCTION

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GRAPHS OF QUADRATIC FUNCTIONS As we’ve already seen, f(x) = x 2 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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GRAPHS OF QUADRATIC FUNCTIONS The standard form of a quadratic function is: f(x) = ax 2 + 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) = x 2 to the following: f(x) = 2x 2 f(x) =.5x 2 f(x) = -.5x 2 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) = x 2 to the following: f(x) = x 2 + 3f(x) = x Vertical shift up Vertical shift down

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GRAPHS OF QUADRATIC FUNCTIONS Now compare f(x) = x 2 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 standard form of a quadratic function f(x) = ax 2 + bx + c is written in the form: 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 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 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 standard form f(x) = ax 2 + bx + c into the 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.

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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 2 - 3g(x) = x 2 - 6x - 1 h(x) = 3x 2 + 6xk(x) = -2x 2 + 6x + 2

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DERIVING THE FORMULA FOR THE VERTEX A formula for the x-coordinate of the vertex can be found by completing the square on the standard form of a quadratic function. f(x) = ax 2 + bx + c

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CHARACTERISTICS OF THE GRAPH OF A QUADRATIC FUNCTION f(x) = ax 2 + bx + c Parabola opens up if a > 0. Parabola opens down 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) = 4x 2 - 8x + 3g(x) = -2x 2 + 8x + 3

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THE X-INTERCEPTS OF A QUADRATIC FUNCTION 1.If the discriminant b 2 – 4ac > 0, the graph of f(x) = ax 2 + bx + c has two distinct x- intercepts and will cross the x-axis twice. 2. If the discriminant b 2 – 4ac = 0, the graph of f(x) = ax 2 + bx + c has one x-intercept and touches the x-axis at its vertex. 3. If the discriminant b 2 – 4ac < 0, the graph of f(x) = ax 2 + bx + c has no x-intercept and will not cross or touch the x-axis.

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FINDING A QUADRATIC FUNCTION Determine the quadratic function whose vertex is (1,- 5) and whose y-intercept is -3.

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CONCLUSION OF SECTION 2.4 CONCLUSION OF SECTION 2.4

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