Problem 15 - Algebra and Trigonometry with Analytic Geometry

Question

Sketch the graph of \(f\). $$f(x)=5\left(\frac{1}{2}\right)^{x}+3$$

Step-by-Step Solution

Answer

The graph is a decreasing exponential function with a y-intercept at (0, 8) and asymptote at y = 3.

Step 1: Identify Function Type

The function given is an exponential function of the form \(f(x) = a \, b^x + c\), where \(a = 5\), \(b = \frac{1}{2}\), and \(c = 3\). The term \(b = \frac{1}{2}\) indicates the function is a decreasing exponential function.

Step 2: Determine Horizontal Asymptote

For the exponential function \(f(x) = 5\left(\frac{1}{2}\right)^{x} + 3\), the horizontal asymptote is determined by the constant \(c\). Thus, \(y = 3\) is the horizontal asymptote.

Step 3: Find Intercept

To find the y-intercept, set \(x = 0\). Then, \(f(0) = 5\left(\frac{1}{2}\right)^{0} + 3 = 5(1) + 3 = 8\). Therefore, the y-intercept is at the point \((0, 8)\).

Step 4: Sketching Few Points

Choose a few x-values to find points on the graph. For example, for \(x = 1:\) \(f(1) = 5\left(\frac{1}{2}\right)^{1} + 3 = \frac{5}{2} + 3 = 5.5\). For \(x = 2:\) \(f(2) = 5\left(\frac{1}{2}\right)^{2} + 3 = \frac{5}{4} + 3 = 4.25\). So, points are \((1, 5.5)\) and \((2, 4.25)\).

Step 5: Sketch the Graph

Start by plotting the y-intercept \((0, 8)\). Next, plot points \((1, 5.5)\) and \((2, 4.25)\). Remember the horizontal asymptote is \(y = 3\), so the graph approaches this line as \(x\to \infty\). Connect these points with a smooth curve decreasing towards the asymptote \(y = 3\) as x increases.

Key Concepts

Decreasing Exponential Function

An exponential function takes the form of \(f(x) = a \, b^x + c\), where \(a > 0\), and \(b\) is a positive constant. When \(0 < b < 1\), the function is classified as a decreasing exponential function. This means that as the value of \(x\) increases, the value of \(b^x\) decreases, leading to a decrease in the overall function value.

In the function \(f(x) = 5\left(\frac{1}{2}\right)^{x} + 3\), \(b = \frac{1}{2}\), which is less than 1. Consequently, the function is decreasing, meaning the graph will slope downwards as \(x\) increases.

Decreasing exponential functions are often used to model processes that decrease over time, such as radioactive decay or cooling of a hot object.

Horizontal Asymptote

The concept of a horizontal asymptote is crucial when understanding the behavior of exponential functions as \(x\) approaches either infinitely large or small values. For any exponential function of the form \(f(x) = a \, b^x + c\), the horizontal asymptote is represented by the constant \(c\).

In our function example, \(f(x) = 5\left(\frac{1}{2}\right)^{x} + 3\), the horizontal asymptote is \(y = 3\).

This indicates that as \(x\) becomes larger and larger, \(f(x)\) gets closer and closer to the value of 3 but never truly reaches it. Graphically, this is depicted by the curve approaching a straight horizontal line, which is the horizontal asymptote.

Y-Intercept

The y-intercept of a function is the point where the graph of the function crosses the y-axis. It is found by setting \(x = 0\) in the function's equation.

For the function \(f(x) = 5\left(\frac{1}{2}\right)^{x} + 3\), by substituting \(x = 0\), we get:

\[f(0) = 5\left(\frac{1}{2}\right)^{0} + 3 = 5(1) + 3 = 8\]

Thus, the y-intercept is the point \((0, 8)\).

This point is critical to graphing, as it provides a starting reference point for plotting the function.

Graph Sketching

Graph sketching involves identifying key points and features to accurately represent a function visually. For exponentials, the process starts by plotting the y-intercept and a few additional points.

In our function, after plotting the y-intercept \((0, 8)\), some chosen points are \((1, 5.5)\) and \((2, 4.25)\). These points provide important markers that help sketch the graph.

• Start with the highest point, the y-intercept.
• Plot additional points like \((1, 5.5)\) and \((2, 4.25)\) to show the decrease in function value.
• Draw a smooth curve through these points that approaches the horizontal asymptote \(y = 3\).

Remember, the curve never actually touches the horizontal asymptote, reaffirming the behavior of a decreasing exponential function.