Problem 9 - College Algebra with Corequisite Support

Question

For the following exercises, graph the function and its reflection about the \(y\) -axis on the same axes, and give the \(y\) -intercept. \(g(x)=-2(0.25)^{x}\)

Step-by-Step Solution

Answer

The function's graph and its reflection have the same y-intercept: (0, -2).

Step 1: Identify the Function

The function given is \( g(x) = -2(0.25)^x \). This is an exponential function with a base of 0.25 and a coefficient of -2.

Step 2: Graph the Function

To graph the function \( g(x) = -2(0.25)^x \), calculate and plot several points. For example:- For \( x = 0 \), \( g(0) = -2(0.25)^0 = -2 \)- For \( x = 1 \), \( g(1) = -2(0.25)^1 = -0.5 \)- For \( x = 2 \), \( g(2) = -2(0.25)^2 = -0.125 \)Plot these points and draw a curve passing through them. The graph will decrease as \( x \) increases from left to right.

Step 3: Reflect the Function About the y-axis

The reflection of the function \( g(x) \) about the \( y \)-axis is given by \( g(-x) = -2(0.25)^{-x} \). Calculate and plot points for this function:- For \( x = 0 \), \( g(-0) = -2(0.25)^{0} = -2 \)- For \( x = 1 \), \( g(-1) = -2(0.25)^{-1} = -8 \)- For \( x = 2 \), \( g(-2) = -2(0.25)^{-2} = -32 \)Plot these points and draw a curve. This graph will appear to increase from left to right.

Step 4: Determine the y-intercept

The \( y \)-intercept of any function is the value when \( x = 0 \). From our calculations:- For \( g(0) = -2 \), the \( y \)-intercept is \( (0, -2) \).- For its reflection \( g(-x) = - 2(0.25)^{-x} \), the \( y \)-intercept remains \( (0, -2) \).

Key Concepts

Reflection

Reflection is an essential concept when you're dealing with graphs of functions. It's like looking at the graph in a mirror placed on a certain axis. Here, we focus on reflecting the exponential function about the y-axis. When you reflect a function across the y-axis, you replace every instance of the variable x with -x in the function's equation.

For example, if you start with the function \( g(x) = -2(0.25)^x \), its reflection about the y-axis is represented as \( g(-x) = -2(0.25)^{-x} \). This transformation modifies the graph's behavior:

• The original function, \( g(x) \), tends to decrease as x increases due to the factor of 0.25, which is less than 1.
• The reflected function, \( g(-x) \), shows increasing behavior as x increases because the base of the exponent \( 0.25 \) is taken to a negative power, reversing the decrease seen in the original function.

Understanding reflections allows you to visualize how a function can change direction while maintaining its shape.

Graphing

Graphing functions helps visualize how they behave across various values. Let's consider the graph of the given function, \( g(x) = -2(0.25)^x \). Observing some points calculated earlier:

• When \( x = 0 \), \( g(0) = -2 \).
• When \( x = 1 \), \( g(1) = -0.5 \).
• When \( x = 2 \), \( g(2) = -0.125 \).

With these points, you can plot a curve. The graph decreases quite rapidly as x increases—this is typical of exponential functions with bases between 0 and 1.

To graph the reflection, use the points for \( g(-x) \):

• When \( x = 0 \), \( g(0) = -2 \).
• When \( x = 1 \), \( g(-1) = -8 \).
• When \( x = 2 \), \( g(-2) = -32 \).

Plotting these points shows a graph that rises instead. This contrasting behavior when compared to \( g(x) \) highlights how reflections affect graph orientations. Both graphs share the y-intercept but diverge in their rise and fall patterns.

y-intercept

The y-intercept is where the graph of a function crosses the y-axis. This is crucial for understanding the starting point of the function's value.

For the function \( g(x) = -2(0.25)^x \), you find the y-intercept by evaluating the function at \( x = 0 \): \[ g(0) = -2(0.25)^0 = -2 \]Hence, the y-intercept of the original function is at the point \((0, -2)\).

Reflecting the function across the y-axis to get \( g(-x) = -2(0.25)^{-x} \) doesn't change its y-intercept. This is because both functions end up with the same value at \( x = 0 \).

• The y-intercept for \( g(-x) \) is also \((0, -2)\).

This commonality shows that while reflections affect the behavior and direction of the graph, they don't alter the y-intercept. Understanding the y-intercept gives a solid point of reference when sketching or analyzing the graph of a function.