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Reed College Department of Mathematics & Statistics

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Worksheet Transformations

1.

Describe how the given equation represents a transformation of the function \(f(x)\text{:}\)
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Answer 1.
\(\text{left 3 units}\)
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Answer 2.
\(\text{right 3 ... 5 units}\)
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3.

Let \(\displaystyle{f(x) = \frac{1}{x}}\text{.}\) Find \(g(x)\text{,}\) the function that is \(f(x)\) shifted down \(4\) units and right \(5\) units.
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\(g(x) =\)
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Hint.
Given a function, \(f(x)\text{,}\) a new function \(g(x) = f(x) +k\) is a vertical shift of the function \(f(x)\text{.}\) If \(k\) is positive the function is shifted up \(k\) units, if \(k\) is negative the function is shifted down \(k\) units.
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Given a function, \(f(x)\text{,}\) a new function \(g(x) = f(x-h)\) is a horizontal shift of the function \(f(x)\text{.}\) If \(h\) is positive the function is shifted right \(h\) units, if \(h\) is negative the function is shifted left \(h\) units.
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Example: \(g(x) = \sqrt{x-4}-5\) is the function \(f(x) =\sqrt{x}\) shifted right 4 units and down 5 units.
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Answer.
\(\frac{1}{x-5}-4\)
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4.

Describe how the given equation represents a transformation of the function \(f(x)\text{:}\)
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Answer 1.
\(\text{vertical stretch}\)
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Answer 2.
\(\text{vertical compression}\)
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5.

Describe how the given equation represents a transformation of the function \(f(x)\text{:}\)
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Answer 1.
\(\text{horizontal compression}\)
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Answer 2.
\(\text{horizontal stretch}\)
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6.

Suppose \(f(t) = 7 t + 5\text{.}\)
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(a) Find \(2 f(t)\) and \(f(2t)\text{.}\)
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\(2f(t) =\)
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\(f(2t) =\)
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(b) Solve the equation \(2 f(t) = f(2t)\text{.}\) Does the equation \(2 f(t) = f(2t)\) have no solution, one solution, or infinitely many solutions?
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If it has no solution, enter NONE. If it has one solution, enter your solution as an equation \(t = a\) for some number \(a\text{.}\) If it has infinitely many solutions, enter an equation \(y = b + m t\) for some numbers \(b\) and \(m\text{.}\)
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Answer 1.
\(2\mathopen{}\left(7t+5\right)\)
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Answer 2.
\(7\cdot 2t+5\)
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Answer 3.
\(\text{no solution}\)
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Answer 4.
\(\text{NONE}\)
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7.

(a) The graph of \(y = f(5 x)\) is the graph of \(y = f(x)\)
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scaled horizontally and
scaled vertically .
(b) The graph of \(y = 2 f(x)\) is the graph of \(y = f(x)\)
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scaled horizontally and
scaled vertically .
(c) The graph of \(y = 4 f(5 x)\) is the graph of \(y = f(x)\)
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scaled horizontally and
scaled vertically .
(d) The graph of \(y = \frac{1}{4} f\left( \frac{x}{5} \right)\) is the graph of \(y = f(x)\)
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scaled horizontally and
scaled vertically .
Answer 1.
\(\text{by compression by a factor of 5}\)
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Answer 2.
\(\text{no vertical scaling}\)
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Answer 3.
\(\text{no horizontal scaling}\)
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Answer 4.
\(\text{by expansion by a factor of 2}\)
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Answer 5.
\(\text{by compression by a factor of 5}\)
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Answer 6.
\(\text{by expansion by a factor of 4}\)
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Answer 7.
\(\text{by expansion by a factor of 5}\)
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Answer 8.
\(\text{by compression by a factor of 4}\)
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8.

Use the two graphs below to answer the following questions.
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(a) The graph of \(y = g(x)\) is the graph of \(y = f(x)\)
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scaled horizontally and
scaled vertically .
(b) Find an equation for \(g(x)\) in terms of the function \(f(x)\text{.}\) For example, \(g(x) = 10 f(9 x + 8) + 7.\)
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\(g(x) =\)
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Graph of \(y = f(x)\)
Graph of \(y = g(x)\)
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Answer 1.
\(\text{by expansion by a factor of 3}\)
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Answer 2.
\(\text{by expansion by a factor of 5}\)
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Answer 3.
\(5f\mathopen{}\left(\frac{x}{3}\right)\)
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9.

Let \(\displaystyle{f(x) = \sqrt{x}}\text{.}\) Find \(g(x)\text{,}\) the function that is \(f(x)\) reflected over the \(x\)-axis and horizontally stretched by a factor of \(2\text{.}\)
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\(g(x) =\)
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Answer.
\(-\sqrt{\frac{1}{2}x}\)
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10.

Describe how the given equation represents a transformation of the function \(f(x)\text{:}\)
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Answer 1.
\(\text{Choice 1}\)
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Answer 2.
\(\text{Choice 3}\)
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12.

(a) Find an equation for \(y = g(x)\) in terms of the function \(y = f(x)\text{.}\) \(g(x) =\) (b) Find an equation for \(y = h(x)\) in terms of the function \(y = f(x)\text{.}\) \(h(x) =\) (c) Find an equation for \(y = j(x)\) in terms of the function \(y = f(x)\text{.}\) \(j(x) =\)
Answer 1.
\(-f\mathopen{}\left(x\right)\)
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Answer 2.
\(f\mathopen{}\left(-x\right)\)
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Answer 3.
\(-f\mathopen{}\left(-x\right)\)
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14.

If the graph of the line \(y = mx + b\) is reflected over the \(y\)-axis, what will be the slope and intercepts of the new graph? (Your answers will depend on the parameters \(b\) and \(m\text{.}\)
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(a) The slope will be
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(b) The \(y\)-intercept will be \(y =\)
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(c) The \(x\)-intercept will be \(x =\)
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Answer 1.
\(-m\)
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Answer 2.
\(b\)
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Answer 3.
\(\frac{b}{m}\)
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Solution.
SOLUTION\(y = mx+b\)\(y\)\(y = m(-x)+b = -mx +b\text{.}\)\(-m\text{,}\)\(y\)\(y = b\text{,}\)\(y\)\(x\)\(x\)\(y=0\text{.}\)\(0 = -m x + b\)\(x\)\(x = b/m\text{.}\)\(x\)
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15.

The graph of \(f(x)=x^3 - 4\) is sketched in red and the graph of \(g(x)\) is sketched in blue. Use the translation rule and \(f(x)=x^3 - 4\) to identify the function \(g(x)\text{;}\)
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\(g(x)=\)
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Answer.
\(-x^{3}-\left(-4\right)\)
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16.

Describe how \(g(x)\) represents a transformation of the function \(f(x)\text{:}\)
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Answer 1.
\(\text{vertical stretch ... -axis}\)
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Answer 2.
\(\text{horizontal ... axis}\)
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17.

Consider the equation \(y = (x+3)^2+2\text{.}\) By hand without using technology, sketch a graph of this equation on paper.
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Which graph A-D below most closely matches the graph you drew?
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A B
C D
(Click on a graph to enlarge it.)
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Consider the equation \(y = -(x+1)^2-2\text{.}\) By hand without using technology, sketch a graph of this equation on paper.
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Which graph A-D below most closely matches the graph you drew?
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A B
C D
(Click on a graph to enlarge it.)
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Answer 1.
\(\text{D}\)
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Answer 2.
\(\text{D}\)
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18.

Suppose \({\left(-13,-2\right)}\) is a point on the graph of \(y=f(x)\text{.}\) What is a point that will be on the graph of \(y=-2f(x)+1\text{?}\)
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Enter a point as \((a,b)\text{,}\) including the parentheses.
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Answer.
\(\left(-13,5\right)\)
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19.

The graph of \(y=f(x)\) is given below (in blue), along with several related graphs (which are in red).
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Note: you can click on the graph to enlarge it.
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For each equation, enter the letter of the corresponding graph.
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\(y= \frac{1}{3} f(x)\)
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\(y=f(x-4)\)
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\(y= -f(x+4)\)
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\(y= f(x)+3\)
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\(y= 2f(x+6)\)
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Answer 1.
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Answer 2.
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Answer 3.
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Answer 4.
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Answer 5.
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20.

(a) Let the graph of \(f(x)\) be given by:
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(Click on graph to enlarge)
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Which of the following is the graph of \(y=2 f(x) - 1\text{?}\)
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A B
C D
(Click on a graph to enlarge it.)
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(b) Let the graph of \(g(x)\) be given by:
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(Click on graph to enlarge)
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Which of the following is the graph of \(y=3 g(x) + 1\text{?}\)
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A B
C D
(Click on a graph to enlarge it.)
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Answer 1.
\(\text{A}\)
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Answer 2.
\(\text{C}\)
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