Sketch the graphs of f(x) = 4^x, f(x) = 4 · 4^x, and f(x) = 1/4 · 4^x on the same set of axes. Then describe the similarities and differences among the graphs.

Steps to Solve

1. Identify the functions to be graphed

The functions to graph are:

• $f(x) = 4^x$
• $f(x) = 4 \cdot 4^x$
• $f(x) = \frac{1}{4} \cdot 4^x$
• $f(x) = 4^{(1/4)x}$

2. Determine key characteristics of each function

• For $f(x) = 4^x$: This is the basic exponential function, which increases rapidly as $x$ increases and approaches $0$ as $x$ decreases.

• For $f(x) = 4 \cdot 4^x$: This function is a vertical stretch of the basic function by a factor of $4$. This means it will have the same general shape but will be higher on the graph.

• For $f(x) = \frac{1}{4} \cdot 4^x$: This is a vertical compression of the basic function, making it lower on the graph by a factor of $\frac{1}{4}$.

• For $f(x) = 4^{(1/4)x}$: This function has a base of $4$ raised to the power of $\frac{1}{4}x$, meaning it will grow slower than $4^x$, depicting a horizontal stretch.

3. Sketch the functions

Plotting the functions on the same set of axes:

• For $f(x) = 4^x$: Start from (0,1), and as $x$ increases, points like (1,4) and (2,16) should be plotted.

• For $f(x) = 4 \cdot 4^x$: It starts higher, at (0,4), and has points like (1,16) and (2,64).

• For $f(x) = \frac{1}{4} \cdot 4^x$: Begin at (0,1) but goes to (1,4) and (2,16), but starts lower for negative $x$ values.

• For $f(x) = 4^{(1/4)x}$: Start at (0,1), but the growth is more gradual; for example, at (4,4) instead of (1,4).

4. Analyze similarities and differences

• Similarities: All functions pass through the point (0,1). They exhibit exponential growth behavior as $x$ increases.

• Differences: The vertical stretches and compressions affect the height. The horizontal stretch or compression (in $f(x) = 4^{(1/4)x}$) affects how quickly the function grows.