In the following exercises you will practice sketching graphs of parabolas by first
finding the vertex. The vertex of a parabola is the high point or low point of the
graph.
We have seen two common forms for equations of parabolas.
Form 1: f (x) = a(x " h) 2 + k
Form 2: f (x) = ax 2 + bx + c
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Let’s take a look at the first form of the parabola, f (x) = a(x " h) 2 + k .
There are some important pieces of information about the parabola that we can get
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instantly from this function. First, if a is positive then the parabola will open up
and if a is negative then the parabola will open down. Second, the vertex of the
parabola is the point (h,k) .
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Now let’s take a look at the second form of the parabola, f (x) = ax 2 + bx + c .
Once again, if a is positive then the parabola will open up and if a is negative then
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the parabola will open down. It takes a little more work to find the vertex of a
parabola of this form. The x-coordinate of the vertex is given by x = "
b
. After
2a
finding the x-coordinate, we can find the y-coordinate by plugging this x-value
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# b # b &&
into our equation. In other words, the vertex is the point %" , f % " (( .
$ 2a $ 2a ''
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2
Example: Find the vertex of the parabola f (x) = ( x "1) + 2 .
This equation is of the form f (x) = a(x " h) 2 + k , with a = 1, h = 1, and k = 2, so
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the vertex is the point (1, 2). Here is a sketch of the graph.
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Example: Find the vertex of the parabola f (x) = "x 2 + 4 x " 5 .
This equation is of the form f (x) = ax 2 + bx + c , with a = "1, b = 4, and c = "5 , so
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the vertex is the point (2, "1). Here is a sketch of the graph.
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Math 1010-001
Worksheet #4
Due 07/14/06
Name:__________________________
In the following exercises, find the vertex of the parabola, then sketch the graph of the
parabola by plotting points on either side of the vertex, or by using the rules for graph
transformations that we discussed in class.
1.) f (x) = (x + 2) 2 + 5
The vertex is __________.
2.) f (x) = 3x 2 " 4 x + 1
The vertex is __________.
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3.) f (x) = 2x 2 + 4 x " 3
The vertex is __________.
2
"
3% 7
4.) f (x) = $ x + ' +
#
4& 5
The vertex is __________.
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5.) f (x) = "(x + 2) 2 + 5
The vertex is __________.
6.) f (x) = "2(x + 1) 2 + 1
The vertex is __________.
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