Problem 24 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the function.\(g(x)=\left(\frac{3}{2}\right)^{x+2}\)

Step-by-Step Solution

Answer

The graph of \(g(x) = (\frac{3}{2})^{x+2}\) is an exponential function that's shifted 2 units to the left. It increases as x increases, and has a horizontal asymptote at \(y=0\). The y-intercept is at \((0, (\frac{3}{2})^2)\).

Step 1: Identify the Base and Exponent

In the function \(g(x) = (\frac{3}{2})^{x+2}\), the base is \(\frac{3}{2}\) and the exponent is \(x+2\). The base \(\frac{3}{2}\) is larger than 1, this means that the graph will be increasing exponential function.

Step 2: Recognize the Shift

With the exponent being \(x+2\), the graph is shifted horizontally. The number attached to x signifies a horizontal shift, and because it’s added to x, it shifts to the left. Therefore, the graph is shifted 2 units to the left.

Step 3: Sketching the graph

First, draw the graph of the function \(g(x) = (\frac{3}{2})^{x}\). Next, move every point in this graph 2 units to the left to account for the +2 in the exponent in \(g(x) = (\frac{3}{2})^{x+2}\). Note that the y-intercept of the graph is at \((0, (\frac{3}{2})^2)\). Also, because the base is greater than 1, the graph increases as x increases. Finally, the line \(y=0\) is a horizontal asymptote for the graph.

Key Concepts

Graph Shifting

When dealing with exponential functions, understanding graph shifting is crucial. In the function given, \(g(x) = \left(\frac{3}{2}\right)^{x+2}\), the exponent \(x+2\) indicates a shift. The transformation involves relocating the entire graph horizontally. Specifically, adding 2 to the exponent results in a 2-unit shift to the left.

Here's why:

• The expression \(x+2\) technically shifts the graph of \(g(x) = \left(\frac{3}{2}\right)^{x}\) by opposite of what the addition would initially suggest.
• Instead of moving to the right, it moves left because horizontal transformations are opposite in signs.

By shifting the graph two units left, you're ensuring the graph starts increasing from further left along the x-axis.

This entirely changes the placement of the graph but keeps its general shape intact.

Horizontal Asymptote

In the world of exponential functions, horizontal asymptotes hold great importance.

A horizontal asymptote is a horizontal line that the graph approaches but never actually reaches. For our function, \(g(x) = \left(\frac{3}{2}\right)^{x+2}\), the horizontal asymptote is \(y = 0\).

Here's why:

• As \(x\) moves towards negative infinity, the function \(\left(\frac{3}{2}\right)^{x+2}\) approaches zero.
• However, due to exponential growth, it never truly becomes zero.

The horizontal asymptote at \(y = 0\) implies that no matter how drastically the graph rises or falls, it will never cross this line.

Understanding this concept is pivotal for correctly imagining the limits and bounds of the graph.

Y-intercept

In graphing, y-intercepts provide a point of reference for visualizing the function. \(g(x) = \left(\frac{3}{2}\right)^{x+2}\) has a y-intercept which can be calculated by evaluating \(g(0)\).

Here's how:

• Set \(x = 0\) in the function: \(g(0) = \left(\frac{3}{2}\right)^{0+2}\).
• This simplifies to \(\left(\frac{3}{2}\right)^{2} = \frac{9}{4}\).

This calculation reveals the y-intercept at \((0, \frac{9}{4})\).

The y-intercept is crucial as it is the starting point from where the graph displays its exponential nature. It anchors the graph, giving you an initial point to plot before applying shifts and other transformations.

Being familiar with y-intercepts helps in setting up your graph quickly and accurately.