Question

Suppose the graph of\({\bf{f}}\)is given. Write equations for the graphs that are obtained from the graph of\({\bf{f}}\)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\({\bf{x}}\)-axis.

(f) Reflect about the\({\bf{y}}\)-axis.

(g) Stretch vertically by a factor of 3.

(h) Shrink vertically by a factor of 3

Short answer

(a)The equation for the graph of\({\rm{f}}\)shifted 3 units upward is

(b) The equation for the graph of\(f\)shifted 3 units downward is

(c) The equation for the graph of\({\rm{f}}\)shifted 3 units to right is

(d) The equation for the graph of\({\rm{f}}\)shifted 3 units to left is

(e) The equation for the reflection of the graph of\({\rm{f}}\)about the x-axis is

(f) The equation for the reflection of the graph of\({\rm{f}}\)about the y-axis is

(g) The equation for the graph of\({\rm{f}}\)stretched vertically by a factor of 3 is

(h) The equation for the graph of\({\rm{f}}\)shrunk vertically by a factor of 3 is

\({\rm{f}}\)

Step-by-step solution

(a) Shift 3 units upward

Consider the graph,\({\rm{y = f}}\left( {\rm{x}} \right)\)

The objective is to write the equation for the graph when the original graph\({\rm{y = f}}\left( {\rm{x}} \right)\)shifted 3 units upward.

If the graph of the function\({\rm{f}}\)is known then for any number\(c > 0\), the graph of the function\({\rm{y = f}}\left( {\rm{x}} \right) + c\)is the graph of the function shifted\(c\)units upward.

In this case,\(c = 3\)

Thus, the equation for the graph of\({\rm{f}}\)shifted 3 units upward is:

(b) Shift 3 units downward

The objective is to write the equation for the graph when the original graph\(y = f\left( x \right)\)shifted 3 units downward.

If the graph of the function\(f\)is known then for any number\(c > 0\), the graph of the function\({\rm{f}}\)shifted\(c\)units downward.

In this case,\(c = 3\)

Thus, the equation for the graph of \(f\)shifted 3 units downward is

(c) Shift 3 units to the right

The objective is to write the equation for the graph when the original graph\({\rm{y = f}}\left( {\rm{x}} \right)\)shifted 3 units right.

If the graph of the function\({\rm{f}}\)is known then for any number\(c > 0\), the graph of the function\({\rm{y = f}}\left( {{\rm{x - c}}} \right)\)is the graph of the function\({\rm{f}}\)shifted\(c\)units right.

In this case,\(c = 3\)

Thus, the equation for the graph of\({\rm{f}}\)shifted 3 units to right is

(d) Shift 3 units to the left

The objective is to write the equation for the graph when the original graph\({\rm{y = f}}\left( {\rm{x}} \right)\)shifted 3 units left.

If the graph of the function\({\rm{f}}\)is known then for any number\(c > 0\), the graph of the function\(y = f\left( {x + c} \right)\) is the graph of the function\({\rm{f}}\)shifted\(c\)units left.

In this case,\(c = 3\)

Thus, the equation for the graph of \({\rm{f}}\)shifted 3 units to left is

(e) Reflect about the \({\bf{x}}\)-axis

The objective is to write the equation for the graph when the original graph\({\rm{y = f}}\left( {\rm{x}} \right)\)reflected about the x-axis.

If the graph of\(y = - f\left( x \right)\)is the reflection if the graph of\({\rm{y = f}}\left( {\rm{x}} \right)\)above the x-axis.

Thus, the equation for the reflection of the graph of \({\rm{f}}\)about the x-axis is:

(f) Reflect about the \({\bf{y}}\)-axis

The objective is to write the equation for the graph when the original graph\({\rm{y = f}}\left( {\rm{x}} \right)\)reflected about the y-axis.

If the graph of\(y = f\left( { - x} \right)\)is the reflection if the graph of\({\rm{y = f}}\left( {\rm{x}} \right)\)above the y-axis.

Thus, the equation for the reflection of the graph of \({\rm{f}}\)about the y-axis is

(g) Stretch vertically by a factor of 3

The objective is to write the equation for the graph when the original graph\({\rm{y = f}}\left( {\rm{x}} \right)\)stretches vertically by a factor of 3.

If the graph of the function\({\rm{f}}\)is known then for any\(c > 0\), the graph of the function\(y = cf\left( x \right)\) is the graph of the function\({\rm{f}}\)stretched vertically by a factor of\(c\).

In this case,\(c = 3\)

Thus, the equation for the graph of \({\rm{f}}\)stretched vertically by a factor of 3 is:

(h) Shrink vertically by a factor of 3

The objective is to write the equation for the graph when the original graph\({\rm{y = f}}\left( {\rm{x}} \right)\)shrinks vertically by a factor of 3.

If the graph of the function\({\rm{f}}\)is known then for any\(c > 0\), the graph of the function\(y = \frac{1}{c}f\left( x \right)\) is the graph of the function\({\rm{f}}\)shrinking vertically by a factor of\(c\).

In this case,\(c = 3\)

Thus, the equation for the graph of \({\rm{f}}\)shrunk vertically by a factor of 3 is: