Problem 20 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The graph of \(f(x)=\left(\frac{3}{2}\right)^{x}\) starts from a height of 0 at x = -∞, crosses the y-axis at y = 1, and then rises to the right towards y = +∞. The plotted points are (-1, 2/3), (0,1), (1, 1.5), and (2, 2.25).

Step 1: Determine Key Points

For exponential functions, one common technique is to choose a few key points to plot. The key points often include x = 0 and a few positive and negative integers. Thus, calculate the values of f(x) at x = -1, 0, 1, and 2.

Step 2: Plot the Key Points

After calculating the function values in Step 1, then plot these points on an x-y plane. The points are (-1, 2/3), (0,1), (1, 1.5), and (2, 2.25).

Step 3: Draw the Graph

Join these plotted points with a smooth curve. Note that because this function is an exponential growth function, the graph will start from a height of 0 at x = -∞, cross the y-axis at y = 1, and then rise to the right towards y = +∞.

Key Concepts

Exponential Growth

Exponential growth is a key concept in mathematics that describes how quantities increase over time. In an exponential function like \( f(x) = \left(\frac{3}{2}\right)^{x} \), the base \( \frac{3}{2} \) is greater than one, illustrating exponential growth. This means that as \( x \) increases, the value of \( f(x) \) rises, showcasing rapid increase over time.
Exponential growth functions are essential as they model real-world phenomena such as population growth, compound interest, and certain areas in science.
To understand why \( f(x) \) represents growth, observe that whenever \( x \) gets larger, the entire quantity \( \left(\frac{3}{2}\right)^{x} \) becomes increasingly large, portraying this growth effect more clearly.

Key Points

In graphing exponential functions, determining key points provides a foundation to construct the graph. These points help us understand the function’s behavior and characteristics.
Key points are specific solutions of the function at select values of \( x \) such as \( x = -1, 0, 1, \) and \( 2 \). The given function yields values at these points:

• When \( x = -1 \), \( f(x) = \frac{2}{3} \)
• When \( x = 0 \), \( f(x) = 1 \)
• When \( x = 1 \), \( f(x) = 1.5 \)
• When \( x = 2 \), \( f(x) = 2.25 \)

These calculated points offer a snapshot into the function's behavior at these critical positions along the graph.
Key points allow us to move forward to plotting and ensure the exponential curve is accurately represented.

Plotting Graphs

Plotting graphs of functions involves representing key points on a plot to visualize the function’s behavior. For our function \( f(x) = \left(\frac{3}{2}\right)^{x} \), after determining the key points, the next step is to plot them.
To begin plotting, draw an x-y plane and mark each point:

• Plot point \((-1, \frac{2}{3})\)
• Plot point \((0, 1)\)
• Plot point \((1, 1.5)\)
• Plot point \((2, 2.25)\)

Graphing these points will reveal the structure of the exponential growth. Observing these points illustrates an increasing trend, as each successive point is higher than the previous one. Plotting correctly is crucial as it lays the groundwork for an accurate depiction of the exponential function.

Smooth Curve

The final step in graphing an exponential function is to connect the plotted key points with a smooth curve. A smooth curve is important because it accurately represents the continuous nature of exponential functions.
When drawing a smooth curve through the points \((-1, \frac{2}{3}), (0, 1), (1, 1.5),\) and \((2, 2.25)\), ensure that the curve smoothly progresses upwards. The right-hand side of the curve should rise more steeply, as the exponential growth is evident as \( x \) becomes larger. The left side should approach a y-value near zero as \( x \) moves to negative infinity.
Always aim for a smooth and flowing line rather than sharp angles or disjointed segments. A well-drawn smooth curve highlights the exponential character of the function and presents an immediate visual understanding of how the exponential relationship manifests.