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Graphing Quadratic Functions 33 22 11 Definitions Rules & Examples Practice Problems

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Definitions A quadratic function is described by an equation of the following forms: Standard Form Vertex Form 2

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Definitions Graphs of a quadratic function Always in the shape of a parabola 3

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Definitions Vertex Minimum or Maximum of the function Minimum is the least value possible of the function Maximum is the greatest value possible of the function Also intercepts the axis of symmetry Axis of Symmetry Vertical line which goes through the exact center of the parabola Y-intercept The point where the parabola “intercepts” the y-axis 4

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Minima & Maxima Parabola opens up (minimum) when “a” is positive Parabola opens down (maximum) when “a” is negative 5

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An easy way to remember Min/Max Positive People 6 Negative People Minimum Maximum SmileFrown

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Find the line of symmetry of y = 3x 2 – 18x + 7 Finding the Line of Symmetry When a quadratic function is in standard form The equation of the line of symmetry is y = ax 2 + bx + c, For example… Using the formula… Thus, the line of symmetry is x = 3.

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Finding the Vertex We know the line of symmetry always goes through the vertex. Thus, the line of symmetry gives us the x – coordinate of the vertex. To find the y – coordinate of the vertex, we need to plug the x – value into the original equation. STEP 1: Find the line of symmetry STEP 2: Plug the x – value into the original equation to find the y value. y = –2x 2 + 8x –3 y = –2(2) 2 + 8(2) –3 y = –2(4)+ 8(2) –3 y = –8+ 16 –3 y = 5 Therefore, the vertex is (2, 5) Note: The y-coordinate of the vertex is also the Minimum/Maximum VALUE of the function

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A Quadratic Function in Standard Form The standard form of a quadratic function is given by y = ax 2 + bx + c There are 3 steps to graphing a parabola in standard form. STEP 1: Find the line of symmetry STEP 2: Find the vertex STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve. Plug in the line of symmetry (x – value) to obtain the y – value of the vertex. MAKE A TABLE using x – values close to the line of symmetry. USE the equation

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STEP 1: Find the line of symmetry Let's Graph ONE! Try … y = 2x 2 – 4x – 1 A Quadratic Function in Standard Form Thus the line of symmetry is x = 1

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Let's Graph ONE! Try … y = 2x 2 – 4x – 1 STEP 2: Find the vertex A Quadratic Function in Standard Form Thus the vertex is (1,–3). Since the x – value of the vertex is given by the line of symmetry, we need to plug in x = 1 to find the y – value of the vertex.

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VERTEX FORM OF QUADRATIC EQUATION y = a(x - h) 2 + k The vertex is (h,k). The axis of symmetry is x = h.

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Graphing Using the Vertex Form Steps Find the Vertex (h,k) & plot Pick 2 values of x, <h Substitute those values into the original function to find the y-coordinate & plot Pick 2 values of x, >h Substitute those values into the original function to find the y-coordinate & plot 13

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GRAPHING A QUADRATIC FUNCTION IN VERTEX FORM (-3,4) (-7,-4) (-1,2) (-5,2) (1,-4) Axis of symmetry x y Example y = -1/2(x + 3) 2 + 4 where a = -1/2, h = -3, k = 4. Since a<0 the parabola opens down. To graph a function, first plot the vertex (h,k) = (-3,4). Draw the axis of symmetry x = -3 Plot two points on one side of it, such as (-1,2) and (1,-4). Use the symmetry to complete the graph.

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INTERCEPT FORM OF QUADRATIC EQUATION y = a(x - p)(x - q) The x intercepts are p and q. The axis of symmetry is halfway between (p,0) and (q,0).

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GRAPHING A QUADRATIC FUNCTION IN INTERCEPT FORM (-2,0) (1,9) (4,0) Axis of symmetry x y Example y = -(x + 2)(x - 4). where a = -1, p = -2, q = 4. Since a<0 the parabola opens down. To graph a function, the x - intercepts occur at (-2,0) and (4,0). Draw the axis of symmetry that lies halfway between these points at x = 1. So, the x - coordinate of the vertex is x = 1 and the y - coordinate of the vertex is: y = -(1 + 2)(1 - 4)= 9.

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