Q71. - Algebra 2

Question

Graph the pair of functions $y=2^x \text{and }y=2^x+3$ on the same screen. Then compare the graphs, listing both similarities and differences in shape, asymptotes, domain, range and y-intercept.

Step-by-Step Solution

Answer

The shape of the two graphs is same. The asymptote of the function $y=2^x$ is $y=0$ and the asymptote of $y=2^x+3$ is $y=3$. The domain of both the functions is the set of all real numbers (R). The range of the function $y=2^x$ is $y>0$, whereas the range of $y=2^x+3$ is $y>3$. The y-intercept of $y=2^x$ is $y=1$ and y-intercept of $y=2^x+3$ is $y=4$.

Step 1. Write down the given equation.

The given functions are $y=2^x \text{and }y=2^x+3$.

Step 2. Sketch the graphs of the functions on the same screen.

The graph of the functions on the same functions on the same screen is shown below.

Step 3. Interpretation from the graph of two functions.

• Shape: The shape of the two graphs are of smooth curve and they are of same shape.
• Asymptotes: The asymptote of the function $y=2^x$ is $y=0$, whereas the asymptote of $y=2^x+3$ is $y=3$.
• Domain: The domain of both the functions is the set of all real numbers (R).
• Range: The range of the function $y=2^x$ is $y>0$, whereas the range of $y=2^x+3$ is $y>3$.
• y-intercept: From the graph it can be seen that the graph of $y=2^x$ cuts y-axis at $\left(0,1\right)$. Hence, y-intercept of $y=2^x$ is $y=1$. And the graph of $y=2^x+3$ cuts y-axis at $\left(0,4\right)$. Hence, y-intercept of $y=2^x+3$ is $y=4$.