Problem 92 - Precalculus

Question

For the following exercises, graph the function and its reflection about the \(x\) -axis on the same axes. $$f(x)=3(0.75)^{x}-1$$

Step-by-Step Solution

Answer

Graph \( f(x) = 3(0.75)^x - 1 \) and its reflection, \( -f(x) = -3(0.75)^x + 1 \), to see one above and one below the x-axis.

Step 1: Understand the Function

The function given is \( f(x) = 3(0.75)^x - 1 \). This is an exponential decay function because the base, \( 0.75 \), is less than 1.

Step 2: Graph the Original Function

To graph \( f(x) = 3(0.75)^x - 1 \), create a table of values. Choose several values for \( x \) (e.g., -2, -1, 0, 1, 2) and calculate corresponding \( f(x) \) values. Plot these points and connect them smoothly to reveal the exponential graph.

Step 3: Determine the Reflection Function

To find the reflection of the function about the \( x \)-axis, substitute \( f(x) \) with \( -f(x) \). For this function, the reflection is \( -f(x) = -(3(0.75)^x - 1) = -3(0.75)^x + 1 \).

Step 4: Graph the Reflected Function

Using the same \( x \)-values as in Step 2, calculate the \( -f(x) \) values using the reflection equation from Step 3. Plot these new points and connect them to visualize the reflection graph.

Step 5: Compare the Two Graphs

Examine both graphs on the same axes. The original function \( f(x) \) should appear above the \( x \)-axis, and its reflection, \( -f(x) \), should appear as a mirror image below the \( x \)-axis.

Key Concepts

Exponential Decay

Exponential decay occurs in mathematical functions where the output decreases as the input increases. For a function to exhibit exponential decay, the base of the exponent must be a fraction less than 1. For instance, the function in our exercise, \( f(x) = 3(0.75)^x - 1 \), is an example of exponential decay. Here, the base is 0.75, which is less than 1, indicating that the function's values decrease as \( x \) increases.

When graphing an exponential decay function:

• The curve will decline rapidly at first and then level off towards its horizontal asymptote.
• In the case of \( f(x) = 3(0.75)^x - 1 \), this horizontal asymptote is at \( y = -1 \).
• The graph never quite touches this line, showing that the function values approach \( -1 \) but never actually reach it.

Understanding exponential decay functions can help comprehend processes like cooling of hot objects or depreciation in economics, where quantities decrease over time at a rate proportional to their current value.

Reflection about x-axis

Reflection about the \( x \)-axis involves flipping a graph over the \( x \)-axis, creating a mirror image effect. To reflect a function about the \( x \)-axis, you multiply the function by -1.

In the exercise, we take \( f(x) = 3(0.75)^x - 1 \) and reflect it by computing \( -f(x) = -(3(0.75)^x - 1) \). Simplifying gives us \( -3(0.75)^x + 1 \). This new function is the reflection and occurs by flipping each \( y \)-coordinate of the original graph to its opposite.

Points which were above the \( x \)-axis in the original \( f(x) \) graph now appear below it in the \( -f(x) \) graph, and vice versa for any points that may have been on the \( x \)-axis. This simple transformation is particularly useful for understanding symmetry in graphs.

Exponential Functions

Exponential functions are fundamental mathematical functions expressed as \( f(x) = ab^x + c \). Here, \( a \) is a constant that stretches or compresses the graph vertically, \( b \) is the base of the exponential term determining growth or decay, and \( c \) shifts the graph vertically up or down.

In our case, \( f(x) = 3(0.75)^x - 1 \) is an exponential decay function:

• \( a = 3 \) dictates the initial steepness and vertical stretch of the graph.
• The base \( b = 0.75 \) reflects the decay rate since it's less than 1.
• \( c = -1 \) shifts the entire graph downwards, setting the horizontal asymptote.

Exponential functions can model a range of real-world applications. When describing populations decreasing over time, the loss of heat in physics, or decline in radioactive substances, exponential functions provide a precise mathematical representation.