GRAPHS OF

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GRAPHS OF QUADRATIC EQUATIONS
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Vocabulary

• Quadratic Equation Equation in the form yax2
  bx c.

• Parabola The general shape of a quadratic
  equation. It is in the form of a U which may
  open upward or downward.

• Vertex The maximum or minimum point of a
  parabola.

• Maximum The highest point (vertex) of a
  parabola when it opens downward.

• Minimum The lowest point (vertex) of a parabola
  when it opens upward.

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Shapes of Graphs
How does the sign of the coefficient of x2 affect
the graph of a parabola?
On your graphing calculator, do the following
1. Press the Y key. 2. Clear any
existing equations by placing the cursor
immediately after the and
pressing CLEAR. 3. Enter 2x2 after the
Y1 by doing the following keystrokes.
2 X,T,? x2 4.
Press GRAPH.
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Up or Down
Repeat using the equation y -2x2.
When the coefficient of x2 is positive, the graph
opens upward. When the coefficient of x2 is
negative, the graph opens downward.
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Narrow or Wide?
How does the value of a in the equation ax2 bx
c affect the graph of the parabola?

• Clear the equations in the Y screen of your
  calculator.
• Enter the equation x2 for Y1.
• Enter the equation 3 x2 for Y2. Choose a
  different type of line for Y2 so that you can
  tell the difference between them.
• Press GRAPH.

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More Narrow or Wide

• Clear the second equation in the Y screen and
  now enter the equation y (1/4)x2.
• Press the GRAPH key and compare the two graphs.

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Summary for ax2

• When a is positive, the parabola opens upward.

• When a is negative, the parabola opens downward.

• When a is larger than 1, the graph will be
  narrower than the graph of x2.

• When a is less than 1, the graph will be wider
  (broader) than the graph of x2.

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Crossing the y-axis
How does the value of c affect the graph of a
parabola when the equation is in the form ax2 c?

• In the Y screen of the graphing calculator,
  enter x2 for Y1.
• Enter x2 3 for Y2.
• Press the GRAPH key.

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Higher or Lower
Now predict what the graph of y x2 5 will
look like.

• Enter x2 for Y1 in the Y screen.
• Enter x2 5 for Y2
• Press GRAPH.

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Left or Right?
What happens to the graph of a parabola when the
equation is in the form (x-h)2 or (xh)2?

• Enter x2 for Y1 in the Y screen.
• Enter (x-3)2 for Y2.
• Press GRAPH.

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Which Way?

• Clear the equation for Y2.
• Enter (x4)2 for Y2.
• Press GRAPH.

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Vertex Summary

• The vertex of the graph of ax2 will be at
  the origin.

• The vertex of the graph of the parabola
  having the equation ax2 c will move up on the
  y-axis by the amount c if cgt0.

• The vertex of the graph of the parabola
  having the equation ax2 c will move down on the
  y-axis by the absolute value of c if clt0.

• The vertex of the graph of the parabola in
  the form (x-h)2 will shift to the right by h
  units on the x-axis.

• The vertex of the graph of the parabola in
  the form (xh)2 will shift to the left by h units
  on the x-axis.

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Practice Problems
Compare the graphs of the following quadratic
equations to each other. Check your work with
your graphing calculator.
1) x2, x2 7, (x 2)2
2) 2x2, x2 6, (1/3)(x-5)2
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Problem 1

• All three graphs have the same shape.
• The vertex of the graph of x2 7 will move
  down 7 on the y-axis.
• the vertex of the graph of (x2)2 will move
  left two on the x-axis.

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Problem 2

• The graph of 2x2 will be the narrowest. The
  graph of (1/3)(x-2)2 will be the broadest.
• The vertex of x2 6 will be shifted up 6
  units on the y-axis compared to the graph of 2x2.
• The vertex of (1/3)(x-2)2 will be shifted
  right two units on the x-axis compared to the
  graph of 2x2.