Concept explainers
Suppose the graph of f is given. Write equations for the graphs that are obtained from the graph of f as follows.
(a) Shift 3 units upward.
(b) Shift 3 units downward.
(c) Shift 3 units to the right.
(d) Shift 3 units to the left.
(e) Reflect about the x-axis.
(f) Reflect about the y-axis.
(g) Stretch vertically by a factor of 3.
(h) Shrink vertically by a factor of 3.
(a)
To write: The equation of the graph which is obtained from the given graph such that the graph shifted 3 units upward.
Answer to Problem 1E
The equation of the graph of f becomes
Explanation of Solution
Let the equation of the graph be
Since the graph is vertically (upward) shifted, add 3 to the
Thus, the equation of the graph of f becomes
(b)
To write: The equation of the graph which is obtained from the given graph such that the graph shifted 3 units downward.
Answer to Problem 1E
The equation of the graph of f becomes
Explanation of Solution
Let the equation of the graph be
Since the graph is vertically (upward) shifted, subtract 3 from the
Thus, the equation of the graph of f becomes
(c)
To write: The equation of the graph which is obtained from the given graph such that the graph shifted 3 units to the right side.
Answer to Problem 1E
The equation of the graph of f becomes
Explanation of Solution
Let the equation of the graph be
Since the graph is horizontally (right side) shifted, subtract 3 from x.
Thus, the equation of the graph of f becomes
(d)
To write: The equation of the graph which is obtained from the given graph such that the graph shifted 3 units to the left side.
Answer to Problem 1E
The equation of the graph of f becomes
Explanation of Solution
Let the equation of the graph be
Since the graph is horizontally (left side) shifted, add 3 to x.
Thus, the equation of the graph of f becomes
(e)
To write: The equation of the graph which is obtained from the given graph such that the graph reflects about the x axis.
Answer to Problem 1E
The equation of the graph of f becomes
Explanation of Solution
Let the equation of the graph be
Since the graph is reflecting about the x-axis, the obtained graph must be an odd function.
Therefore, substitute
Thus, the equation of the graph of f becomes
(f)
To write: The equation of the graph which is obtained from the given graph such that the graph reflects about the y axis.
Answer to Problem 1E
The equation of the graph of f becomes
Explanation of Solution
Let the equation of the graph be
Since the graph reflects about the x-axis, the obtained graph must be an even function.
Therefore, substitute
Thus, the equation of the graph of f becomes
(g)
To write: The equation of the graph which is obtained from the given graph such that the graph stretched vertically by a factor of 3.
Answer to Problem 1E
The equation of the graph of f becomes
Explanation of Solution
Let the equation of the graph be
Since the graph is stretched vertically by a factor of 3, multiply 3 to the
Thus, the equation of the graph of f becomes
(h)
To write: The equation of the graph which is obtained from the given graph such that the graph shrunk vertically by a factor of 3.
Answer to Problem 1E
The equation of the graph of f becomes
Explanation of Solution
Let the equation of the graph be
Since the graph is shrunk vertically by a factor of 3, divide 3 to the
Thus, the equation of the graph of f becomes
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Single Variable Calculus: Concepts and Contexts, Enhanced Edition
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