Question

Graph the pair of functions $y={\left(\frac{1}{5}\right)}^x \text{and}y={\left(\frac{1}{5}\right)}^{x-2}$on the same screen. Then compare the graphs, listing both similarities and differences in shape, asymptotes, domain, range and y-intercept.

Short answer

The shape of the two graphs is same. The asymptote of the both functions is horizontal $y=0$. The domain of both the functions is the set of all real numbers (R). The range of the both functions is $y>0$. The y-intercept of $y={\left(\frac{1}{5}\right)}^x$ is $y=1$ and the y-intercept of $y={\left(\frac{1}{5}\right)}^{x-2}$ is $y=25$.

Step-by-step solution

Step 1. Write down the given equation.

The given functions are $y={\left(\frac{1}{5}\right)}^x \text{and}y={\left(\frac{1}{5}\right)}^{x-2}$.

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 $y={\left(\frac{1}{5}\right)}^x$ is horizontal $y=0$, whereas the asymptote of $y={\left(\frac{1}{5}\right)}^{x-2}$ is $y=0$.
• Domain:The domain of both the functions is the set of all real numbers (R).
• Range: The range of the function $y={\left(\frac{1}{5}\right)}^x$ is $y>0$, and the range of $y={\left(\frac{1}{5}\right)}^{x-2}$ is $y>0$.
• y-intercept:From the graph it can be seen that they-intercept of $y={\left(\frac{1}{5}\right)}^x$ is $y=1$ and the y-intercept of $y={\left(\frac{1}{5}\right)}^{x-2}$ is $y=25$.