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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 Arctic Poppy Satellite Dishes The Golden Gate Bridge Trajectory Headlights

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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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**y = ax2 + bx + c Standard Form Examples**

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**Minimum point is also called a vertex. y = ax2 + 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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**Maximum point is also called a vertex. y = -ax2 + 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 = ax2 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. X Y (X,Y)

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(X,Y) -2, 8 -1,2 0,0 1,2 2,8

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Standard Form y = ax2+c Examples

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Ex. X Y (X,Y)

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(X,Y) -2, 11 -1,5 0,3 1,5 2,11

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Check It Out! Example 2b Graph the quadratic function. y = –3x2 + 1 Make a table of values. Choose values of x and use them to find values of y. x –2 –1 1 2 y 1 –2 –11 Graph the points. Then connect the points with a smooth curve.

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Ex. X Y (X,Y)

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(X,Y) -2,-8 -1,-2 0,0 1,-2 2,-8

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Check It Out! Example 2a Graph each quadratic function. y = x2 + 2 Make a table of values. Choose values of x and use them to find values of y. x –2 –1 1 2 y 2 3 6 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. Write the function in the form y = ax2 + bx + c by solving for y. Add to both sides. Identify the value of a. Since a > 0, the parabola opens upward.

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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 – 3x2 Write the function in the form y = ax2 + bx + c. y = –3x2 + 5x a = –3 Identify the value of a. Since a < 0, the parabola opens downward.

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Check It Out! Example 3a Tell whether the graph of each quadratic function opens upward or downward. Explain. f(x) = –4x2 – x + 1 f(x) = –4x2 – 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 = 2x2 – 5. 2. Graph y = 1.5x2. 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. (5, –4) maximum; –4 D: all real numbers; R: y ≤ –4

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Graph quadratic functions.

Graph quadratic functions.

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