Question

Use a graphing utility to graph \(f(x)=e^{x}\) and the given function in the same viewing window. How are the two graphs related? (a) \(g(x)=e^{x-2}\) (b) \(h(x)=-\frac{1}{2} e^{x}\) (c) \(q(x)=e^{-x}+3\)

Short answer

Comparisons: The graph of \(g(x)\) is the same as that of \(f(x)\) but shifted 2 units to the right. The graph of \(h(x)\) is a reflection of \(f(x)\) across the x-axis, also compression by a factor of 1/2. The graph of \(q(x)\) is a reflection of \(f(x)\) across the y-axis and shifted upwards by 3 units.

Step-by-step solution

Graphing the Base Function

Start by graphing the basic function \(f(x)=e^{x}\). This graph should serve as the reference for other graphs.

Graphing \(g(x)=e^{x-2}\)

Next, graph the function \(g(x)=e^{x-2}\). Notice any changes from base function. In this case, the graph will shift 2 units to the right because of the subtraction inside the exponent.

Comparing \(f(x)\) and \(g(x)\)

Visually compare the graph of \(f(x)=e^{x}\) with the graph of \(g(x)=e^{x-2}\). The two graphs are identical, except \(g(x)\) is shifted two units to the right.

Graphing \(h(x)=-\frac{1}{2} e^{x}\)

Graph the function \(h(x)=-\frac{1}{2} e^{x}\). In this case, there is a multiplication by -1/2. Therefore, the graph will flip across the x-axis and will also be vertically compressed.

Comparing \(f(x)\) and \(h(x)\)

Visually compare the graph of \(f(x)=e^{x}\) with the graph of \(h(x)=-\frac{1}{2} e^{x}\). The graph of \(h(x)\) is a reflection of \(f(x)\) across the x-axis, and is vertically compressed by a factor of 1/2.

Graphing \(q(x)=e^{-x}+3\)

Graph the function \(q(x)=e^{-x}+3\). In this case, there is a multiplication inside the exponent by -1 and an addition outside it by 3. Thus the graph will flip across the y-axis and shift upward by 3 units.

Comparing \(f(x)\) and \(q(x)\)

Visually compare the graph of \(f(x)=e^{x}\) with the graph of \(q(x)=e^{-x}+3\). The graph of \(q(x)\) is a reflection of \(f(x)\) across the y-axis and is shifted upwards by 3 units.