2.4.1: Exercises 2.4
- Page ID
- 63332
Terms and Concepts
Exercise \(\PageIndex{1}\)
Why do changes on the inside of the function produce horizontal changes?
- Answer
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Answers will vary, but should include the idea that changes on the inside are changes to the inputs, which are on the horizontal axis.
Exercise \(\PageIndex{2}\)
Why do changes on the outside of the function produce vertical changes?
- Answer
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Answers will vary, but should include the idea that applying changes on the outside of the function affects the outputs, which are the y values (height of the graph).
Exercise \(\PageIndex{3}\)
In graphing the function \(g(x)=2 \ln{(x)} + 4\), which transformation should you apply first?
- Answer
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The vertical stretch
Exercise \(\PageIndex{4}\)
In graphing the function \(f(x)=(2x-1)^3\), which transformation should you apply first?
- Answer
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The horizontal shift
Exercise \(\PageIndex{5}\)
In graphing the function \(h(t)=3^{t+4}\), what is the base function and how is it being transformed?
- Answer
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The base function is \(3^t\) and it is being shifted 4 units to the left
Problems
Graph each of the functions in exercises \(\PageIndex{6}\) - \(\PageIndex{10}\).
Exercise \(\PageIndex{6}\)
\(f(x)=-x^2+1\)
- Answer
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Exercise \(\PageIndex{7}\)
\(f(x) = \left\{\begin{array}{cc} 2x+8 & x \leq -1 \\ -x+7 & x>-1 \end{array}\right.\)
- Answer
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Exercise \(\PageIndex{8}\)
\(f(x) = \left\{\begin{array}{cc} -x^2 & x < 0 \\ (x-1)^2 & 0 \leq x <3 \end{array}\right.\)
- Answer
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Exercise \(\PageIndex{9}\)
\(f(x) = e^x+1\)
- Answer
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Exercise \(\PageIndex{10}\)
\(f(x) = \left\{\begin{array}{cc} \sin{(x)} & x < \pi \\ \cos{(x)} & x> \pi \end{array}\right.\)
- Answer
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In exercises \(\PageIndex{11}\) - \(\PageIndex{15}\), graph and write an equation for each of the described functions.
Exercise \(\PageIndex{11}\)
The result of shifting \(g(x)=x^2\) up three units and to the left two units
- Answer
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\(h(x)=(x+2)^2+3\)
Exercise \(\PageIndex{12}\)
Any even degree polynomial that is positive for \(-2 \leq x \leq 4\).
- Answer
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Answers will vary, but an example is
\(h(x)=x^2+1\)
Exercise \(\PageIndex{13}\)
The result of shifting \(f(\theta) = -2\theta +3\) down two units and right 5 units.
- Answer
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\(g(\theta) = -2(\theta-5) + 3 -2=-2\theta + 11\)
Exercise \(\PageIndex{14}\)
The piecewise function that consists of \(t^2\), shifted down one unit for \(t \leq -2\) and of the line with a slope of 3 and a y-intercept of 3 for \(t>-2\).
- Answer
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Add texts here. Do not delete this text first.
\(f(t) = \left\{\begin{array}{cc} t^2-1 & t \leq 2 \\ 3t+3 & -2<x \end{array}\right.\)
Exercise \(\PageIndex{15}\)
The line with a slope of \(\frac{2}{3}\) that passes through the point \((1,f(2))\), where \(f(x)=x^2-1\).
- Answer
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\(y-3 = \frac{2}{3} (x-1)\), or \(y=\frac{2}{3} x + \frac{7}{3}\)
In exercises \(\PageIndex{16}\) - \(\PageIndex{18}\), factor the given function, and graph the function.
Exercise \(\PageIndex{16}\)
\(b(x)=x^3+6x^2+12x+8\)
- Answer
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\(b(x)=(x+2)^3\);
Exercise \(\PageIndex{17}\)
\(y(t) = t^2-6t+9\)
- Answer
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\(y(t)=(t-3)^2\);
Exercise \(\PageIndex{18}\)
\(f(x) = x^2+4x+4\)
- Answer
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\(f(x)=(x+2)^2\);
For each of
- \(f(x)=x^{2}-3x,\)
- \(\eta(\theta )=\cos(\theta ),\) and
- \(g(w)=3^{w}-w^{3},\)
write the equation for the new function that results from the transofmration(s) stated in exercises \(\PageIndex{19}\) - \(\PageIndex{24}\).
Exercise \(\PageIndex{19}\)
Add exercises text here.
- Answer
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Shift up 3 units
\(x^2-3x+3\);
\(\cos{(\theta)} + 3\);
\(3^w - w^3 + 3\)
Exercise \(\PageIndex{20}\)
Shift right 2 units
- Answer
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\((x-2)^2-3(x-2)\);
\(\cos{(\theta-2)}\);
\(3^{w-2} - (w-2)^3\)
Exercise \(\PageIndex{21}\)
Shift down 2 units and left 1 unit
- Answer
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\((x+1)^2-3(x+1) -2\);
\(\cos{(\theta+1)} -2\);
\(3^{w+1} - (w+1)^3 -2\)
Exercise \(\PageIndex{22}\)
Shift down \(\pi\) units and right \(e\) units
- Answer
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\((x-e)^2-3(x-e) - \pi\);
\(\cos{(\theta-e)} - \pi\);
\(3^{w-e} - (w-e)^3 - \pi\)
Exercise \(\PageIndex{23}\)
Flip across the x-axis
- Answer
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\((-x)^2-3(-x)\);
\(\cos{(-\theta)}\);
\(3^{-w} - (-w)^3\)
Exercise \(\PageIndex{24}\)
Flip across the y-axis
- Answer
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\(-(x^2-3x)\);
\(-\cos{(\theta)}\);
\(-(3^{-w} - w^3)\)
Answer each question in exercises \(\PageIndex{25}\) - \(\PageIndex{26}\) using the provided graphs.
Exercise \(\PageIndex{25}\)
Based on the shape of the graph of \(f(x)\) shown, below,
- could \(f(x)\) be an even polynomial? Why or why not?
- could \(f(x)\) be an odd polynomial? Why or why not?
- could \(f(x)\) be an exponential function? Why or why not?
- could \(f(x)\) be a trigonometric function? Why or why not?
- Answer
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- no; it is not symmetric about the y-axis
- yes; it has rotational symmetry
- no; it does not have a horizontal asymptote
- no; it does not have a periodic (repeating) pattern
Exercise \(\PageIndex{26}\)
Based on the shape of the graph of \(g(x)\) shown, below,
- could \(g(x)\) be an even polynomial? Why or why not?
- could \(g(x)\) be an odd polynomial? Why or why not?
- could \(g(x)\) be an exponential function? Why or why not?
- could \(g(x)\) be a trigonometric function? Why or why not?
- Answer
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- no; it is not symmetric about the y-axis
- no; it does not have rotational symmetry
- yes; it does not have a horizontal asymptote on one side and grows without bound on the other
- no; it does not have a periodic (repeating) pattern