11.8E: Exercises
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Practice Makes Perfect
Exercise \(\PageIndex{23}\) Graph Quadratic Functions of the Form \(f(x)=x^{2}=k\)
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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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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Exercise \(\PageIndex{24}\) Graph Quadratic Functions of the Form \(f(x)=x^{2}=k\)
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.
Exercise \(\PageIndex{25}\) Graph Quadratic Functions of the Form \(f(x)=(x-h)^{2}\)
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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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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Exercise \(\PageIndex{26}\) Graph Quadratic Functions of the Form \(f(x)=(x-h)^{2}\)
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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3.
5.
Exercise \(\PageIndex{27}\) Graph Quadratic Functions of the Form \(f(x)=(x-h)^{2}\)
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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3.
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7.
Exercise \(\PageIndex{28}\) Graph Quadratic Functions of the Form \(f(x)=ax^{2}\)
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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Exercise \(\PageIndex{29}\) Graph Quadratic Functions Using Transformations
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. \(f(x)=3(x+1)^{2}-4\)
Exercise \(\PageIndex{30}\) Graph Quadratic Functions Using Transformations
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.
- \(f(x)=-(x-4)^{2}+0\)
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- \(f(x)=-(x+2)^{2}+6\)
11.
- \(f(x)=5(x-1)^{2}+3\)
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- \(f(x)=2(x-1)^{2}-1\)
15.
- \(f(x)=-2(x-2)^{2}-2\)
Exercise \(\PageIndex{31}\) Graph Quadratic Functions Using Transformations
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.
- \(f(x)=-(x-1)^{2}-3\)
Exercise \(\PageIndex{32}\) Matching
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
Exercise \(\PageIndex{33}\) Find a Quadratic Function from its Graph
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. \(f(x)=2(x-1)^{2}-3\)
Exercise \(\PageIndex{34}\) Writing Exercise
- 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?