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Exploring Quadratic Graphs
Objective:To graph quadratic
functions.
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ParabolaThe 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 = ax2 + bx + c
Examples 2 2 25 7 3y xy xx y x
http://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 = 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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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?
2 6 4x x
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Steps1. 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)
22y x
(0,0)00
22x
12
12
22(1) 2 2 (1,2)22(2) 8 8 (2,8)
22( 1) 2
22( 2) 8
28
( 1,2)( 2,8)
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(X,Y)-2, 8-1,20,01,22,8
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Ex. X Y (X,Y)
22 +3 y x
(0,3)30
22 +3 x
12
12
22(1) 3 5 5 (1,5)22(2) 3 11 11 (2,11)
22( 1) 3 5
22( 2) 3 11 511
( 1,5)( 2,11)
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(X,Y)-2, 11-1,50,31,5
2,11
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Graph the quadratic function.
Check It Out! Example 2b
y = –3x2 + 1
x
–2
–1
0
1
2
y
1
–2
–11
–2
–11
Make a table of values.Choose values of x anduse them to find valuesof y.
Graph the points. Then connect the points with a smooth curve.
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Ex. X Y (X,Y)
2( ) 2 f x x
(0,0)00
22 x
12
12
22(1) 2 2 (1, 2)22(2) 8 8 (2, 8)
22( 1) 2
22( 2) 8 28
( 1, 2) ( 2, 8)
22(0) 0
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(X,Y)-2,-8-1,-20,01,-22,-8
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Graph each quadratic function.
Check It Out! Example 2a
y = x2 + 2
x
–2
–1
0
1
2
y
2
3
3
6
6
Make a table of values.Choose values of x anduse them to find valuesof 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 = ax2 + 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 – 3x2
y = –3x2 + 5x
a = –3 Identify the value of a.
Since a < 0, the parabola opens downward.
Write the function in the form y = ax2 + 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) = –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.
D: all real numbers;R: y ≤ –4
maximum; –4
(5, –4)