Question

Describe the transformation of \(f\) represented by \(g\). Then graph each function. \(f(x)=4^x, g(x)=4^x-8\)

Short answer

The transformation of \(f(x)=4^x\), resulting in \(g(x)=4^x-8\), is a vertical shift of the graph downwards by 8 units. Thus, graphing \(g(x)\) involves graphing \(f(x)=4^x\) and then shifting the entire graph downwards by 8 units.

Step-by-step solution

Identify the base function and transformation.

Look at the functions given. The function \(f(x)=4^x\) is the base or parent function. The function \(g(x)=4^x-8\) can be seen as the parent function \(f(x)=4^x\) minus 8. Thus, the transformation from \(f(x)\) to \(g(x)\) is a vertical shift downward by 8 units.

Graph the base function.

Start by graphing \(f(x)\) on a graph. The function \(4^x\) is an exponential function, so it will start close to 0 for negative values of x, cross the y-axis at 1, and then increase exponentially for positive values of x.

Perform and graph the transformation.

Next, perform the transformation on each point of the function \(f\) by shifting it downwards by 8 units. After shifting all points, draw the graph of \(g(x)\), taking care to maintain the shape of the original \(f(x)\).

Key concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In the given exercise, our base function is \(f(x) = 4^x\). Here, the base is 4 and the exponent is \(x\). This type of function shows rapid growth or decay:

• For positive values of \(x\), the function grows rapidly as \(x\) increases.
• When \(x\) is negative, the output becomes positive but less than 1, approaching zero but never reaching it.

Exponential functions are widely used to model real-world scenarios such as population growth, radioactive decay, and interest calculations. Their graphs have a distinctive shape:

• They pass through the point (0, 1) because anything to the power of zero is 1.
• They display a horizontal asymptote, usually along the \(x\)-axis, where the graph never touches or crosses.

These properties make exponential functions essential in many studies of growth processes.

Vertical Shift

A vertical shift is a transformation that moves a graph up or down without changing its shape. In the context of the exercise, we apply a vertical shift to the function \(f(x) = 4^x\) to get \(g(x) = 4^x - 8\).Here, you subtract 8 from the output of the original function:

• This means every point on the graph of \(f(x)\) is moved down by 8 units.
• As a result, the new graph, \(g(x)\), shares the same shape as \(f(x)\), but it is positioned 8 units lower on the graph.

Understanding vertical shifts helps us modify the position of graphs while maintaining their inherent properties such as rate of growth or decay. Vertical shifts are common in adjusting functions for various practical applications, aligning them with data or adjusting their starting points for easier interpretation.

Graphing Functions

Graphing functions is a crucial skill for visualizing and understanding their behavior. To graph the base function \(f(x) = 4^x\), start by identifying key points:

• When \(x = 0\), \(f(x) = 1\).
• For negative \(x\), \(f(x)\) approaches zero but never actually reaches it, depicting a horizontal asymptote along the \(x\)-axis.
• For positive \(x\), the function grows exponentially.

Once the base graph is plotted, apply any transformations. In this case, move each point down by 8 units to graph \(g(x) = 4^x - 8\). When graphing, maintain accuracy by checking points at different \(x\)-values to ensure the transformation is consistent. This process allows us to see how exponential functions and their transformations look visually, making it easier to interpret mathematical ideas in a more concrete form.