Problem 558 - Precalculus

Question

Graph the function \(f(x)=5(0.5)^{-x}\) and its reflection across the \(y\) -axis on the same axes, and give the \(y\) -intercept.

Step-by-Step Solution

Answer

The \( y \)-intercept is \( (0, 5) \), and both functions intersect the \( y \)-axis at this point.

Step 1: Understand the Function

The given function is an exponential function of the form \( f(x) = 5(0.5)^{-x} \). Here, \(5\) is the vertical stretch factor, \(0.5\) is the base of the exponential, and \(-x\) in the exponent indicates that this function decreases as \(x\) increases.

Step 2: Determine the Y-intercept

To find the \( y \)-intercept, set \( x = 0 \) in the function. This gives \( f(0) = 5(0.5)^{-0} = 5 \). Thus, the \( y \)-intercept of the function is \( (0, 5) \).

Step 3: Reflect the Function Across the Y-axis

The reflection of a function \( f(x) \) across the \( y \)-axis is given by \( f(-x) \). For our function, this becomes \( 5(0.5)^{-(-x)} = 5(0.5)^{x} \). This is the reflected function.

Step 4: Graph the Original Function

Plot the function \( f(x) = 5(0.5)^{-x} \). This function will decrease exponentially from the \( y \)-intercept \( (0, 5) \) as \( x \) increases.

Step 5: Graph the Reflected Function

Now graph the reflected function \( f(x) = 5(0.5)^{x} \) on the same axes. It starts at the same \( y \)-intercept \( (0, 5) \) but increases exponentially as \( x \) increases.

Key Concepts

Graphing Functions

Understanding how to graph functions is vital when dealing with different types of mathematical expressions, especially exponential functions. For the function \( f(x) = 5(0.5)^{-x} \), it is an exponential function because it is in the form of \( a(b)^{-x} \), where \( a = 5 \) and \( b = 0.5 \). The base \( b \) is less than 1, which signifies a decaying function. In this specific case, since there's a negative exponent, the function decreases as \( x \) increases.

When graphing this function, you would start by plotting the \( y \)-intercept, which is the point where the graph intersects the \( y \)-axis. Then, as you increase the value of \( x \), you can see the plot settle towards the x-axis in a descending manner. It's essential while graphing to choose several points by plugging in different \( x \) values to get corresponding \( y \) values, ensuring the graph is accurate and reflects the proper nature of the function.

Y-Intercept

The \( y \)-intercept is the point where a graph crosses the \( y \)-axis. To find it, you simply set \( x \) to 0 in the function and solve for \( f(x) \). This helps to find where the function starts on the \( y \)-axis.

For the function \( f(x) = 5(0.5)^{-x} \), when \( x = 0 \), plug it in to get \( f(0) = 5(0.5)^{-0} \). This simplifies to \( f(0) = 5 \times 1 = 5 \). Thus, the \( y \)-intercept is \( (0, 5) \). This point is crucial because it gives a starting point for graphing and helps determine the initial value of the exponential function on the \( y \)-axis.

Reflection Across the Y-axis

Reflecting a function across the \( y \)-axis changes its orientation on the coordinate plane. When a function \( f(x) \) is reflected across the \( y \)-axis, you replace \( x \) with \( -x \) to get the reflected function \( f(-x) \).

For the function \( f(x) = 5(0.5)^{-x} \), the reflection across the \( y \)-axis forms the function \( f(-x) = 5(0.5)^{x} \). This reflected function will have the same \( y \)-intercept as the original function but displays an increasing behavior instead of decreasing. It moves upwards from the \( y \)-intercept, diverging away from the \( x \)-axis, which is a classic characteristic of an increasing exponential function.

This reflection is immensely helpful for understanding the behavior of functions and their symmetry properties, providing insights into how transformations affect function graphs.