Problem 77 - Precalculus

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 y-intercept is at (0, -2); graph the function and its reflection.

Step 1: Understanding the function

The given function is \( g(x) = -2(0.25)^x \). This is an exponential function where the base is 0.25 and the coefficient is -2. The base being less than 1 indicates this function will decay as \( x \) increases.

Step 2: Reflect the function across the y-axis

To find the reflection of the function \( g(x) = -2(0.25)^x \) across the y-axis, replace \( x \) with \( -x \). This gives us the reflected function \( g(-x) = -2(0.25)^{-x} \).

Step 3: Graph the original function

Create a table of values for \( g(x) = -2(0.25)^x \) by choosing values for \( x \) (such as -2, -1, 0, 1, 2) and calculate corresponding \( g(x) \) values. Plot these points on a graph and connect them smoothly as the function is exponential.

Step 4: Graph the reflected function

Similarly, create a table of values for \( g(-x) = -2(0.25)^{-x} \) using the same \( x \) values but replacing \( x \) with \( -x \). Plot these points on the same graph as the original function and connect them.

Step 5: Find the y-intercept

The \( y \)-intercept occurs where \( x = 0 \). Substitute \( x = 0 \) into the original function \( g(x) \): \( g(0) = -2(0.25)^0 = -2(1) = -2 \). Thus, the \( y \)-intercept is at \( (0, -2) \).

Key Concepts

Reflection Across Y-Axis

In mathematics, reflecting a function across the y-axis involves creating a mirror image of its graph. To perform this reflection for a function such as \( g(x) = -2(0.25)^x \), simply replace every occurrence of \( x \) with \( -x \). This results in the reflected function \( g(-x) = -2(0.25)^{-x} \).
When you reflect across the y-axis, the transformation aims to show how the function behaves oppositely on both sides of the y-axis.

The visual concept is straightforward:

• The original function decays as it moves rightward (increasing \( x \) values).
• The reflected function, however, decays as it moves leftward since \( x \) is replaced by \( -x \), flipping the base of the exponent.

This reflection helps in understanding symmetry in the graphical representation of functions, showing inverse behavior across the y-axis.

Function Graphing

Graphing functions like \( g(x) = -2(0.25)^x \) involves plotting points based on the function's formula. Here's how you can approach this:
1. **Choose values for \( x \):** Start with a range of numbers such as -2, -1, 0, 1, and 2. This gives a balanced view of the function on both sides of the y-axis.
2. **Calculate \( g(x) \) for each \( x \):** Plug each x-value into the function to find corresponding y-values. For example, when \( x = 0, g(x) = -2(0.25)^0 = -2 \). The point is (0, -2).
3. **Plot and connect the points:** Each calculated pair is a point on the graph. Plot each on a coordinate plane (x, y). Connect them smoothly, keeping in mind that the function is exponential and should curve.

For the reflected function \( g(-x) = -2(0.25)^{-x} \) perform similar steps but use \( -x \) in calculations. This means inverting the x-coordinates for calculations, then plotting in the same plane, revealing symmetry between both graphs.

Y-Intercept Calculation

The y-intercept of a function is where the graph crosses the y-axis. This occurs when \( x = 0 \). To find the y-intercept of \( g(x) = -2(0.25)^x \), substitute \( x = 0 \) into the function:
\[ g(0) = -2(0.25)^0 \]
Any number raised to the power of zero is 1, so \( 0.25^0 = 1 \). This gives:
\[ g(0) = -2 \times 1 = -2 \]
This calculation shows that the y-intercept is at the point (0, -2). It represents the starting point on the vertical axis for the graph of the function, providing a key reference for sketching the function's behavior.

Understanding the y-intercept is crucial as it impacts how the graph starts and provides a fixed location along the y-axis regardless of transformations applied to the function itself.