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Exploring Quadratic Graphs Objective: To graph quadratic functions.

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Parabola The graph of any quadratic function. It is a kind of curve.

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Where are parabolas seen in the real world? The Golden Gate Bridge Satellite Dishes Headlights Trajectory The Arctic Poppy

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Why is the parabola important? Suspension Bridges use a parabolic design to evenly distribute the weight of the entire bridge to the supporting columns.

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Why is the parabola important? The Satellite Dish uses a parabolic shape to ensure that no matter where on the dish surface the satellite signal strikes, it is always reflected to the receiver.

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Why is the parabola important? A car’s Headlights, and common flashlights, use parabolic mirrors to project the light from the bulb into a tight beam, directing the light straight out from the car, or flashlight.

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Standard Form y = ax 2 + bx + c Examples http://www.mathwarehouse.com/quadratic/parabola/interactive- parabola.phphttp://www.mathwarehouse.com/quadratic/parabola/interactive- parabola.php or http://www-groups.dcs.st- and.ac.uk/~history/Java/Parabola.html

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y = ax 2 + bx + c Positive “a” values mean the parabola will open upwards and will have a minimum. point Minimum point is also called a vertex.

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y = -ax 2 + bx + c Negative “a” values mean the parabola will open downwards and will have a maximum point Maximum point is also called a vertex.

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Will the graph open up or down?

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Standard Form y = ax 2 Examples

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Steps 1. Draw a table and insert vertex of (0,0). 2. Choose two numbers greater than the x coordinate and two numbers less. 3. Solve for Y Graph

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Ex. XY(X,Y)

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-2, 8 -1,2 0,0 1,2 2,8

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Standard Form y = ax 2 +c Examples

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Ex. XY(X,Y)

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-2, 11 -1,5 0,3 1,5 2,11

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Graph the quadratic function. Check It Out! Example 2b y = – 3x 2 + 1 x –2 –2 –1 –1 0 1 2 y 1 –2–2 – 11 –2–2 Make a table of values. Choose values of x and use them to find values of y. Graph the points. Then connect the points with a smooth curve.

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Ex. XY(X,Y)

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-2,-8 -1,-2 0,0 1,-2 2,-8

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Graph each quadratic function. Check It Out! Example 2a y = x 2 + 2 x –2 –2 –1 –1 0 1 2 y 2 3 3 6 6 Make a table of values. Choose values of x and use them to find values of y. Graph the points. Then connect the points with a smooth curve.

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Additional Example 3A: Identifying the Direction of a Parabola Tell whether the graph of the quadratic function opens upward or downward. Explain. Since a > 0, the parabola opens upward. Identify the value of a. Write the function in the form y = ax 2 + bx + c by solving for y. Add to both sides.

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Additional Example 3B: Identifying the Direction of a Parabola Tell whether the graph of the quadratic function opens upward or downward. Explain. y = 5x – 3x 2 y = – 3x 2 + 5x a = –3 Identify the value of a. Since a < 0, the parabola opens downward. Write the function in the form y = ax 2 + bx + c.

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Check It Out! Example 3a Tell whether the graph of each quadratic function opens upward or downward. Explain. f(x) = –4x 2 – x + 1 Identify the value of a. a = –4 Since a < 0 the parabola opens downward.

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Lesson Quiz: Part I 1. Without graphing, tell whether (3, 12) is on the graph of y = 2x 2 – 5. 2. Graph y = 1.5x 2. no

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Lesson Quiz: Part II Use the graph for Problems 3-5. 3. Identify the vertex. 4. Does the function have a minimum or maximum? What is it? 5. Find the domain and range. D: all real numbers; R: y ≤ –4 maximum; – 4 (5, –4)

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