Problem 9 - College Algebra

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 functions intersect the \( y \)-axis at (0, -2).

Step 1: Identify the Original Function

The original function given is \( g(x) = -2 (0.25)^x \). This is an exponential function characterized by a base less than 1, which typically represents a decaying function.

Step 2: Graph the Original Function

Plot the graph of \( g(x) = -2 (0.25)^x \). Start by substituting a few \( x \) values into the function to find corresponding \( g(x) \) values. For example: \( g(0) = -2 \), \( g(1) = -0.5 \), \( g(2) = -0.125 \). Plot these points and connect them smoothly, noting that the graph will decline rapidly as \( x \) increases.

Step 3: Determine the Reflection

To find the reflection of the function about the \( y \)-axis, we consider \( g(-x) \), turning our function into \( g(-x) = -2 (0.25)^{-x} \). Simplify it to \( g(-x) = -2 \times 4^x \), which is consistent with the property of negative exponents transforming the base.

Step 4: Graph the Reflected Function

Graph \( g(-x) = -2 \times 4^x \) by substituting some \( x \) values similar to the original function. For example: \( g(-0) = -2 \), \( g(-1) = -8 \), \( g(-2) = -128 \). This function will reflect symmetrically over the \( y \)-axis compared to the original function.

Step 5: Identify the Intersection with the Y-axis

The \( y \)-intercept is where the graph of a function intersects the \( y \)-axis. For both the original and reflected functions, this occurs at \( x = 0 \). Substituting \( x = 0 \) into either function, we find \( g(0) = -2 \), so the \( y \)-intercept is at the point \( (0, -2) \).

Key Concepts

Graphing Functions

Graphing functions involves plotting points on the coordinate plane to visualize the behavior of equations. In this case, we're working with the exponential function \( g(x) = -2(0.25)^x \). Exponential functions have a unique trait where the rate of change is proportional to the function's current value, leading to rapid increases or decreases.
To graph the function, we select several \( x \) values and compute the corresponding \( g(x) \) values. For instance:

• For \( x = 0 \): \( g(0) = -2 \)
• For \( x = 1 \): \( g(1) = -0.5 \)
• For \( x = 2 \): \( g(2) = -0.125 \)

Plot these points on the graph. As \( x \) increases, the function rapidly approaches zero, showcasing the decaying nature of exponential functions with a base less than one. With these points plotted, we draw a smooth curve through them, illustrating the overall shape of the function.

Reflecting Functions

Reflecting a function over the \( y \)-axis involves altering the function's domain. This is done by replacing \( x \) with \( -x \) in the function's equation. Applying this to \( g(x) = -2(0.25)^x \) gives us \( g(-x) = -2 (0.25)^{-x} \), which simplifies using the property of negative exponents to \( g(-x) = -2 \times 4^x \).
The reflection creates a new function that's reversed horizontally compared to the original. To graph it, we substitute \( x \) values into \( g(-x) \) like before:

• For \( x = 0 \): \( g(-0) = -2 \)
• For \( x = 1 \): \( g(-1) = -8 \)
• For \( x = 2 \): \( g(-2) = -128 \)

These points reveal how the reflected graph displays a swiftly increasing behavior, asymmetrically mirroring the original function. The reflection showcases the symmetrical nature of exponential graphs across the \( y \)-axis.

Y-Intercept

The \( y \)-intercept of a function is where the graph crosses the \( y \)-axis, occurring at \( x = 0 \). It's a crucial feature that provides instant insights into the function's behavior. For both the original and reflected versions of our function, we find:

• Substituting \( x = 0 \) into \( g(x) = -2(0.25)^x \), results in \( g(0) = -2 \)

Therefore, the \( y \)-intercept of the function is at the point \((0, -2)\). This consistent intercept means both the original and the reflected functions intersect the \( y \)-axis at the same point, establishing the origin as a point of intersection. Recognizing the \( y \)-intercept helps in quickly anchoring the graph during sketching and understanding the base level from which the function begins its exponential behavior.