Chapter 1.3, Problem 1E

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.

To determine

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 $y=f\left(x\right)+3$.

Explanation of Solution

Let the equation of the graph be $y=f\left(x\right)$.

Since the graph is vertically (upward) shifted, add 3 to the $f\left(x\right)$.

Thus, the equation of the graph of f becomes $y=f\left(x\right)+3$.

To determine

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 $y=f\left(x\right)-3$.

Explanation of Solution

Let the equation of the graph be $y=f\left(x\right)$.

Since the graph is vertically (upward) shifted, subtract 3 from the $f\left(x\right)$.

Thus, the equation of the graph of f becomes $y=f\left(x\right)-3$.

To determine

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 $y=f\left(x-3\right)$.

Explanation of Solution

Let the equation of the graph be $y=f\left(x\right)$.

Since the graph is horizontally (right side) shifted, subtract 3 from x.

Thus, the equation of the graph of f becomes $y=f\left(x-3\right)$.

To determine

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 $y=f\left(x+3\right)$.

Explanation of Solution

Let the equation of the graph be $y=f\left(x\right)$.

Since the graph is horizontally (left side) shifted, add 3 to x.

Thus, the equation of the graph of f becomes $y=f\left(x+3\right)$.

To determine

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 $y=-f\left(x\right)$.

Explanation of Solution

Let the equation of the graph be $y=f\left(x\right)$.

Since the graph is reflecting about the x-axis, the obtained graph must be an odd function.

Therefore, substitute $f\left(x\right)$ by $-f\left(x\right)$.

Thus, the equation of the graph of f becomes $y=-f\left(x\right)$.

To determine

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 $y=f\left(-x\right)$.

Explanation of Solution

Let the equation of the graph be $y=f\left(x\right)$.

Since the graph reflects about the x-axis, the obtained graph must be an even function.

Therefore, substitute $f\left(x\right)$ by $f\left(-x\right)$.

Thus, the equation of the graph of f becomes $y=f\left(-x\right)$.

To determine

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 $y=3f\left(x\right)$.

Explanation of Solution

Let the equation of the graph be $y=f\left(x\right)$.

Since the graph is stretched vertically by a factor of 3, multiply 3 to the $f\left(x\right)$.

Thus, the equation of the graph of f becomes $y=3f\left(x\right)$.

To determine

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 $y=\frac{f\left(x\right)}{3}$.

Explanation of Solution

Let the equation of the graph be $y=f\left(x\right)$.

Since the graph is shrunk vertically by a factor of 3, divide 3 to the $f\left(x\right)$.

Thus, the equation of the graph of f becomes $y=\frac{f\left(x\right)}{3}$.