Question

The graph of a function f is given. Sketch the graph of the following transformations of f.

(a) $y=f(x+1)$

(b) $y=f(-x)$

(c) $y=f(x-2)$

(d) $y=f\left(x\right)-2$

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

(f) $y=2f\left(x\right)$

Short answer

Graph of the functions are

(a)$y=f\left(x+1\right)$

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

(c)$y=f\left(x-2\right)$

(d)$y=f\left(x\right)-2$

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

(f)$y=2f\left(x\right)$

Step-by-step solution

Part a. Step 1. Horizontal shifts of graph.

The value of $f(x-c)$at $x$is same as the value of $f\left(x\right)$at $x-c$since $x-c$is $c$units to the left of x, it follows that the graph of $y=f(x-c)$is just the graph of $y=f\left(x\right)$shifted to the right c units.

Similarly, the graph of $y=f(x+c)$is the graph of $y=f\left(x\right)$shifted to the left c units.

Part a. Step 2. Given graph.

The graph of f is

Part a. Step 3. Graph of Solution.

The graph of$y=f\left(x+1\right)$

Part b. Step 1. Reflecting of graph.

To graph $y=-f\left(x\right)$, reflect the graph $y=f\left(x\right)$ in the x-axis.

To graph $y=f\left(-x\right)$, reflect the graph $y=f\left(x\right)$ in the y-axis.

Part b. Step 2. Given graph.

The graph of f is

Part b. Step 3. Graph of Solution.

The graph of $y=f\left(-x\right)$

Part c. Step 1. Horizontal shifts of graph.

The value of $f(x-c)$at $x$is same as the value of $f\left(x\right)$at $x-c$since $x-c$is $c$units to the left of x, it follows that the graph of $y=f(x-c)$is just the graph of $y=f\left(x\right)$shifted to the right c units.

Similarly, the graph of $y=f(x+c)$is the graph of $y=f\left(x\right)$shifted to the left c units.

Part c. Step 2. Given graph.

The graph of f is

Part c. Step 3. Graph of Solution.

The graph of $y=f\left(x-2\right)$

Part d. Step 1. Vertical shifts of graph.

Adding a constant to a function shifts its graph vertically upward or downward if it is positive or negative respectively.

Part d. Step 2. Given graph.

The graph of f is

Part d. Step 3. Graph of Solution.

The graph of $y=f\left(x\right)-2$

Part e. Step 1. Reflecting of graph.

To graph $y=-f\left(x\right)$, reflect the graph $y=f\left(x\right)$in the x-axis.

To graph $y=f\left(-x\right)$, reflect the graph $y=f\left(x\right)$ in the y-axis.

Part e. Step 2. Given graph.

The graph of f is

Part e. Step 3. Graph of Solution.

The graph of $y=-f\left(x\right)$

Part f. Step 1. Vertical stretching and shrinking.

To graph $y=cf\left(x\right)$

If $c>1$, stretch the graph of $y=f\left(x\right)$vertically by a factor of c.

If $0<c<1$, shrink the graph of $y=f\left(x\right)$vertically by a factor of c.

Part f. Step 2. Given graph.

The graph of f is

Part f. Step 3. Graph of Solution.

The graph is $y=2f\left(x\right)$