Chapter 4.1, Problem 3E

1. (a) To obtain the graph of g(x) = 2^x − 1, we start with the graph of f(x) = 2^x and shift it _____ (upward/downward) 1 unit.
2. (b) To obtain the graph of h(x) = 2^x−1, we start with the graph of f(x) = 2^x and shift it to the _____ (left/right) 1 unit.

To determine

To fill: The direction of graph for $g\left(x\right)=2^x-1$ with the graph $f\left(x\right)=2^x$.

Answer to Problem 3E

The graph of $g\left(x\right)=2^x-1$ shifts downward by 1 unit with the graph of $f\left(x\right)=2^x$.

Explanation of Solution

The given exponential function of $f\left(x\right)$ is, $f\left(x\right)=2^x$.

Add a constant c in function $f\left(x\right)$,

$f\left(x\right)=2^x+c$ (1)

Add a constant to a function shifts its graph horizontally.

If c is positive then the graph will shifts upward by c unit at all points.

If c is negative then the graph will shifts downward by c unit at all points.

The given function of $g\left(x\right)$ is,

$g\left(x\right)=2^x-1$ (2)

Compare equation (1) and (2),

$c=-1$.

Since the value of c is $-1$, the graph of $g\left(x\right)$ shifts downward by 1 units with the graph of $f\left(x\right)$.

Hence, the graph of $g\left(x\right)=2^x-1$ shifts downward by 1 unit with the graph of $f\left(x\right)=2^x$.

To determine

To fill: The direction of graph for $h\left(x\right)=2^{x-1}$ with the graph $f\left(x\right)=2^x$.

Answer to Problem 3E

The graph of $h\left(x\right)=2^{x-1}$ shifts right by 1 unit with the graph of $f\left(x\right)=2^x$.

Explanation of Solution

The given exponential function of $f\left(x\right)$ is, $f\left(x\right)=2^x$.

Add a constant c in function $f\left(x\right)$,

$f\left(x\right)=2^{x+c}$ (1)

Add a constant to a function shifts its graph vertically.

If c is positive then the graph will shift to the left by c unit at all points.

If c is negative then the graph will shift to the right by c unit at all points.

The given function of $g\left(x\right)$ is,

$g\left(x\right)=2^{x-1}$ (2)

Compare equation (1) and (2),

$c=-1$.

Since the value of c is $-1$, the graph of $g\left(x\right)$ shifts right by 1 unit with the graph of $f\left(x\right)$.

Hence, the graph of $g\left(x\right)=2^x-1$ shifts right by 1 unit with the graph of $f\left(x\right)=2^x$.