9.8E: Exercises
Practice Makes Perfect
In the following exercises,
- Graph the quadratic functions on the same rectangular coordinate system
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Describe what effect adding a constant, \(k\), to the function has on the basic parabola.
- \(f(x)=x^{2}, g(x)=x^{2}+4, \text { and } h(x)=x^{2}-4\)
- \(f(x)=x^{2}, g(x)=x^{2}+7, \text { and } h(x)=x^{2}-7\)
- Answer
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1.
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Figure 9.7.71 - The graph of \(g(x)=x^{2}+4\) is the same as the graph of \(f(x)=x^{2}\) but shifted up \(4\) units. The graph of \(h(x)=x^{2}-4\) is the same as the graph of \(f(x)=x^{2}\) but shift down \(4\) units.
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In the following exercises, graph each function using a vertical shift.
- \(f(x)=x^{2}+3\)
- \(f(x)=x^{2}-7\)
- \(g(x)=x^{2}+2\)
- \(g(x)=x^{2}+5\)
- \(h(x)=x^{2}-4\)
- \(h(x)=x^{2}-5\)
- Answer
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1.
3.
5.
In the following exercises,
- Graph the quadratic functions on the same rectangular coordinate system
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Describe what effect adding a constant, \(h\), inside the parentheses has
- \(f(x)=x^{2}, g(x)=(x-3)^{2}, \text { and } h(x)=(x+3)^{2}\)
- \(f(x)=x^{2}, g(x)=(x+4)^{2}, \text { and } h(x)=(x-4)^{2}\)
- Answer
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1.
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Figure 9.7.75 - The graph of \(g(x)=(x−3)^{2}\) is the same as the graph of \(f(x)=x^{2}\) but shifted right \(3\) units. The graph of \(h(x)=(x+3)^{2}\) is the same as the graph of \(f(x)=x^{2}\) but shifted left \(3\) units.
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In the following exercises, graph each function using a horizontal shift.
- \(f(x)=(x-2)^{2}\)
- \(f(x)=(x-1)^{2}\)
- \(f(x)=(x+5)^{2}\)
- \(f(x)=(x+3)^{2}\)
- \(f(x)=(x-5)^{2}\)
- \(f(x)=(x+2)^{2}\)
- Answer
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1.
3.
5.
In the following exercises, graph each function using transformations.
- \(f(x)=(x+2)^{2}+1\)
- \(f(x)=(x+4)^{2}+2\)
- \(f(x)=(x-1)^{2}+5\)
- \(f(x)=(x-3)^{2}+4\)
- \(f(x)=(x+3)^{2}-1\)
- \(f(x)=(x+5)^{2}-2\)
- \(f(x)=(x-4)^{2}-3\)
- \(f(x)=(x-6)^{2}-2\)
- Answer
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1.
3.
5.
7.
In the following exercises, graph each function.
- \(f(x)=-2 x^{2}\)
- \(f(x)=4 x^{2}\)
- \(f(x)=-4 x^{2}\)
- \(f(x)=-x^{2}\)
- \(f(x)=\frac{1}{2} x^{2}\)
- \(f(x)=\frac{1}{3} x^{2}\)
- \(f(x)=\frac{1}{4} x^{2}\)
- \(f(x)=-\frac{1}{2} x^{2}\)
- Answer
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1.
3.
5.
7.
In the following exercises, rewrite each function in the \(f(x)=a(x−h)^{2}+k\) form by completing the square.
- \(f(x)=-3 x^{2}-12 x-5\)
- \(f(x)=2 x^{2}-12 x+7\)
- \(f(x)=3 x^{2}+6 x-1\)
- \(f(x)=-4 x^{2}-16 x-9\)
- Answer
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1. \(f(x)=-3(x+2)^{2}+7\)
3. \(f(x)=3(x+1)^{2}-4\)
In the following exercises,
- Rewrite each function in \(f(x)=a(x−h)^{2}+k\) form
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Graph it by using transformations
- \(f(x)=x^{2}+6 x+5\)
- \((x)=x^{2}+4 x-12\)
- \(f(x)=x^{2}+4 x-12\)
- \(f(x)=x^{2}-6 x+8\)
- \(f(x)=x^{2}-6 x+15\)
- \(f(x)=x^{2}+8 x+10\)
- \(f(x)=-x^{2}+8 x-16\)
- \(f(x)=-x^{2}+2 x-7\)
- \(f(x)=-x^{2}-4 x+2\)
- \(f(x)=-x^{2}+4 x-5\)
- \(f(x)=5 x^{2}-10 x+8\)
- \(f(x)=3 x^{2}+18 x+20\)
- \(f(x)=2 x^{2}-4 x+1\)
- \(f(x)=3 x^{2}-6 x-1\)
- \(f(x)=-2 x^{2}+8 x-10\)
- \(f(x)=-3 x^{2}+6 x+1\)
- Answer
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1.
- f(x)=(x+3)^{2}-4
3.
- \(f(x)=(x+2)^{2}-1\)
5.
- \(f(x)=(x-3)^{2}+6\)
7.
- \(f(x)=-(x-4)^{2}+0\)
9.
- \(f(x)=-(x+2)^{2}+6\)
11.
- \(f(x)=5(x-1)^{2}+3\)
13.
- \(f(x)=2(x-1)^{2}-1\)
15.
- \(f(x)=-2(x-2)^{2}-2\)
In the following exercises,
- Rewrite each function in \(f(x)=a(x−h)^{2}+k\) form
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Graph it using properties
- \(f(x)=2 x^{2}+4 x+6\)
- \(f(x)=3 x^{2}-12 x+7\)
- \(f(x)=-x^{2}+2 x-4\)
- \(f(x)=-2 x^{2}-4 x-5\)
- Answer
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1.
- \(f(x)=2(x+1)^{2}+4\)
3.
- \(f(x)=-(x-1)^{2}-3\)
In the following exercises, match the graphs to one of the following functions:
- \(f(x)=x^{2}+4\)
- \(f(x)=x^{2}-4\)
- \(f(x)=(x+4)^{2}\)
- \(f(x)=(x-4)^{2}\)
- \(f(x)=(x+4)^{2}-4\)
- \(f(x)=(x+4)^{2}+4\)
- \(f(x)=(x-4)^{2}-4\)
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\(f(x)=(x-4)^{2}+4\)
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Figure 9.7.97 -
Figure 9.7.98 -
Figure 9.7.99 -
Figure 9.7.100 -
Figure 9.7.101 -
Figure 9.7.102 -
Figure 9.7.103 -
Figure 9.7.104
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- Answer
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1. c
3. e
5. d
7. g
In the following exercises, write the quadratic function in \(f(x)=a(x−h)^{2}+k\) form whose graph is shown.
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Figure 9.7.105 -
Figure 9.7.106 -
Figure 9.7.107 -
Figure 9.7.108
- Answer
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1. \(f(x)=(x+1)^{2}-5\)
3. \(f(x)=2(x-1)^{2}-3\)
- Graph the quadratic function \(f(x)=x^{2}+4x+5\) first using the properties as we did in the last section and then graph it using transformations. Which method do you prefer? Why?
- Graph the quadratic function \(f(x)=2x^{2}−4x−3\) first using the properties as we did in the last section and then graph it using transformations. Which method do you prefer? Why?
- Answer
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1. Answers may vary.
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
b. After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?