Problem 78 - 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. $$h(x)=6(1.75)^{-x}$$

Step-by-Step Solution

Answer

The \( y \)-intercept is 6.

Step 1: Understand the function

The given function is \( h(x) = 6(1.75)^{-x} \). This is an exponential decay function, as indicated by the negative exponent. The base of the exponent is greater than 1, and the negative exponent implies decay as \( x \) increases.

Step 2: Reflect the function

To find the reflection of the function about the \( y \)-axis, replace \( x \) with \( -x \). The reflection function is \( h(-x) = 6(1.75)^{x} \). This changes the function to an exponential growth function.

Step 3: Find the y-intercept of h(x)

To find the \( y \)-intercept, evaluate \( h(x) \) when \( x = 0 \). This gives \( h(0) = 6(1.75)^{0} = 6 \times 1 = 6 \). So, the \( y \)-intercept of \( h(x) \) is 6.

Step 4: Graph the functions

Graph \( h(x) = 6(1.75)^{-x} \), which is a decreasing function starting from \( y = 6 \) at \( x = 0 \), and approaches 0 as \( x \) increases. Graph the reflection \( h(-x) = 6(1.75)^{x} \), which is an increasing function starting from \( y = 6 \) at \( x = 0 \), increasing as \( x \) increases. Both graphs should clearly show the transformation about the \( y \)-axis and pass through the \( y \)-intercept at 6.

Key Concepts

Exponential Decay

Exponential decay refers to a situation where quantities decrease rapidly at a consistent rate. In the function \(h(x) = 6(1.75)^{-x}\), the exponent \(-x\) is negative, which is a key indicator of decay. This means as \(x\) gets larger, the value of \(h(x)\) becomes smaller. Here's why:

• When \(x\) increases, \(1.75\) raised to a larger negative power becomes a smaller fraction.
• This results in \(h(x)\) decreasing towards zero, though it never actually reaches zero.

The base \(1.75\) is important here, because it is greater than one, which generally suggests growth. But the negative exponent flips this natural tendency, making the function decay instead. This is distinct from other decay scenarios where the base itself is less than one.

Reflection About the Y-Axis

Reflection about the y-axis is an interesting graph transformation. It involves flipping a given function across the y-axis, changing positive \(x\) coordinates to negative, and vice versa. For \(h(x) = 6(1.75)^{-x}\), the reflection is achieved through the transformation of \(x\) to \(-x\). This results in the function \(h(-x) = 6(1.75)^{x}\). Here are some key points about the reflection:

• The original function \(h(x)\) starts high and decays as \(x\) increases, reflecting about the y-axis leads to \(h(-x)\), which rises instead of falls for increasing positive \(x\).
• Graphically, this operation means all points that were on the right of y-axis in the original function now mirror to the left, creating a symmetrical graph that shows exponential growth.

Y-Intercept

The y-intercept represents the point where a graph crosses the y-axis. Every function that can touch the y-axis has a y-intercept, calculated by evaluating the function at \(x = 0\). For \(h(x) = 6(1.75)^{-x}\), the y-intercept can be easily calculated:

• When \(x = 0\), the expression \(1.75^{-0}\) simplifies to 1.
• This leaves us with \(h(0) = 6 \times 1 = 6\).

So, the y-intercept for this function is 6. It is the same for the reflected function \(h(-x)\) because, when \(x = 0\), the function simplifies the same way. This consistent value highlights the initial value of the function at the starting point of the graph.