Calculus: Early Transcendentals, 8th Edition
Authors: James Stewart
ISBN-13: 978-1285741550
See our solution for Question 1E from Chapter 1.3 from Stewart's Calculus, 8th Edition.
Problem 1E
Chapter:
Problem:
Suppose the graph of f is given. Write equations for the graphs that are obtained from the graph of f as follows.
Step-by-Step Solution
Given information
We are given with some graph $f$. We have to find the equations of grpah upon shifting the curve.
Step 1: (a) Shift 3 units upwards
Given graph is \[y = f\left( x \right)\]When graph is shifted 3 units upwards, all the y coordinates are increased by 3 units. So the new graph is: \[\begin{array}{l}y' = y + 3\\ = f\left( x \right) + 3\end{array}\]Therefore, the equation of the new graph is
\[y' = f\left( x \right) + 3\]
Step 2: (b) Shift 3 units downwards
Given graph is \[y = f\left( x \right)\]When graph is shifted 3 units downwards, all the y coordinates are decreased by 3 units. So the new graph is: \[\begin{array}{l}y' = y - 3\\ = f\left( x \right) - 3\end{array}\]Therefore, the equation of the new graph is
\[y' = f\left( x \right) - 3\]
Step 3: (c) Shift 3 units right
Given graph is \[y = f\left( x \right)\]When graph is shifted 3 units right, all the x coordinates are increased by 3 units. So the new graph is: Therefore, the equation of the new graph is
\[y' = f\left( x - 3 \right) \]
Step 4: (d) Shift 3 units left
Given graph is \[y = f\left( x \right)\]When graph is shifted 3 units right, all the x coordinates are decreased by 3 units. So the new graph is: Therefore, the equation of the new graph is
\[y' = f\left( x + 3 \right) \]
Step 5: (e) Reflect about x axis
Given graph is \[y = f\left( x \right)\]The y coordinates are changed to negative y coordinates. The reflection about x axis is given by:
\[y' = - f\left( x \right)\]
Step 6: (f) Reflect about y axis
Given graph is \[y = f\left( x \right)\]The x coordinates are changed to negative x coordinates. The reflection about y axis is given by:
\[y' = f\left( - x \right)\]
Step 7: (g) Stretch vertically by a factor of 3
Given graph is \[y = f\left( x \right)\]During vertically stretching the function, the y coordinates are multiplied by a factor. So the new graph is
\[y' = 3f\left( x \right)\]
Step 8: (h) Shrink vertically by a factor of 3
Given graph is \[y = f\left( x \right)\]During vertically Shrinking the function, the y coordinates are divided by a factor. So the new graph is
\[y' = \dfrac{{f\left( x \right)}}{3}\]
We are given with some graph $f$. We have to find the equations of grpah upon shifting the curve.
Step 1: (a) Shift 3 units upwards
Given graph is \[y = f\left( x \right)\]When graph is shifted 3 units upwards, all the y coordinates are increased by 3 units. So the new graph is: \[\begin{array}{l}y' = y + 3\\ = f\left( x \right) + 3\end{array}\]Therefore, the equation of the new graph is
\[y' = f\left( x \right) + 3\]
Step 2: (b) Shift 3 units downwards
Given graph is \[y = f\left( x \right)\]When graph is shifted 3 units downwards, all the y coordinates are decreased by 3 units. So the new graph is: \[\begin{array}{l}y' = y - 3\\ = f\left( x \right) - 3\end{array}\]Therefore, the equation of the new graph is
\[y' = f\left( x \right) - 3\]
Step 3: (c) Shift 3 units right
Given graph is \[y = f\left( x \right)\]When graph is shifted 3 units right, all the x coordinates are increased by 3 units. So the new graph is: Therefore, the equation of the new graph is
\[y' = f\left( x - 3 \right) \]
Step 4: (d) Shift 3 units left
Given graph is \[y = f\left( x \right)\]When graph is shifted 3 units right, all the x coordinates are decreased by 3 units. So the new graph is: Therefore, the equation of the new graph is
\[y' = f\left( x + 3 \right) \]
Step 5: (e) Reflect about x axis
Given graph is \[y = f\left( x \right)\]The y coordinates are changed to negative y coordinates. The reflection about x axis is given by:
\[y' = - f\left( x \right)\]
Step 6: (f) Reflect about y axis
Given graph is \[y = f\left( x \right)\]The x coordinates are changed to negative x coordinates. The reflection about y axis is given by:
\[y' = f\left( - x \right)\]
Step 7: (g) Stretch vertically by a factor of 3
Given graph is \[y = f\left( x \right)\]During vertically stretching the function, the y coordinates are multiplied by a factor. So the new graph is
\[y' = 3f\left( x \right)\]
Step 8: (h) Shrink vertically by a factor of 3
Given graph is \[y = f\left( x \right)\]During vertically Shrinking the function, the y coordinates are divided by a factor. So the new graph is
\[y' = \dfrac{{f\left( x \right)}}{3}\]