Question

compare the graph of each function to the graph of $f(x) = 2^x$.1. $g(x) = 2^x + 1$2. $p(x) = 2^{x-1}$3. $j(x) = 2^x - 4$4. $g(x) = 2^{x+1}$5. compare the function represented by the graph of $g(x) = 2^x - 3$ to the function represented by the table.| $x$ | $f(x)$ || ---- | ---- || $-2$ | $1.25$ |

Explanation:

Step1: Identify parent function

Parent function: $f(x)=2^x$

Step2: Analyze vertical shift (1)

For $g(x)=2^x + 1$, apply $h(x)=f(x)+k$.
Here $k=1>0$, so shift up 1 unit.

Step3: Analyze horizontal shift (2)

For $p(x)=2^{x-1}$, apply $h(x)=f(x-h)$.
Here $h=1>0$, so shift right 1 unit.

Step4: Analyze vertical shift (3)

For $j(x)=2^x - 4$, apply $h(x)=f(x)+k$.
Here $k=-4<0$, so shift down 4 units.

Step5: Analyze horizontal shift (4)

For $g(x)=2^{x+1}$, apply $h(x)=f(x-h)$.
Rewrite as $2^{x-(-1)}$, $h=-1<0$, so shift left 1 unit.

Step6: Find table's function (5)

For $x=-2$, $f(-2)=2^{-2}=\frac{1}{4}=0.25$, table value is $1.25=0.25+1$.
For $x$ general, table function is $2^x +1$. Compare to $g(x)=2^x-3$:
$g(x)=(2^x+1)-4$, so shift table's graph down 4 units.

Answer:

1. The graph of $g(x)=2^x + 1$ is the graph of $f(x)=2^x$ shifted 1 unit upward.
2. The graph of $p(x)=2^{x-1}$ is the graph of $f(x)=2^x$ shifted 1 unit to the right.
3. The graph of $j(x)=2^x - 4$ is the graph of $f(x)=2^x$ shifted 4 units downward.
4. The graph of $g(x)=2^{x+1}$ is the graph of $f(x)=2^x$ shifted 1 unit to the left.
5. The function from the table is $2^x + 1$, so the graph of $g(x)=2^x - 3$ is the graph of the table's function shifted 4 units downward.