Problem 17 - Algebra and Trigonometry with Analytic Geometry

Question

Sketch the graph of \(f\). $$f(x)=-\left(\frac{1}{2}\right)^{x}+4$$

Step-by-Step Solution

Answer

The graph is a downward-sloping exponential decay, approaching 4 as an asymptote, with a y-intercept at (0, 3).

Step 1: Identify the Function Type

The function given is \( f(x) = -\left(\frac{1}{2}\right)^x + 4 \), which is an exponential function. Exponential functions have the general form \( f(x) = ab^x + c \). In this case, \( a = -1 \) (reflects over the x-axis), \( b = \frac{1}{2} \) (decaying exponential), and \( c = 4 \) (vertical shift upwards by 4 units).

Step 2: Determine the Asymptote

Exponential functions have a horizontal asymptote. For this function, the horizontal asymptote is \( y = c = 4 \). As \( x \to \infty \), \( f(x) \to 4 \) from below, due to the negative sign in front of the exponential.

Step 3: Find the Y-intercept

To find the y-intercept, set \( x = 0 \). Calculate \( f(0) = -\left(\frac{1}{2}\right)^0 + 4 = -1 + 4 = 3 \). Thus, the y-intercept is \( (0, 3) \).

Step 4: Calculate a Few Key Points

Calculate values for a few points to better sketch the curve. For \( x = -1 \), \( f(-1) = -2 + 4 = 2 \). For \( x = 1 \), \( f(1) = -\frac{1}{2} + 4 = 3.5 \). These help to visualize how the curve behaves.

Step 5: Sketch the Graph

Plot the y-intercept (0, 3) and other points found: (-1, 2) and (1, 3.5). The graph should approach the asymptote \( y = 4 \) without crossing it, starting from below. The curve slopes downward due to the negative coefficient of \( \left(\frac{1}{2}\right)^x \). The overall shape is an exponential decay reflected over the x-axis with a vertical shift.

Key Concepts

Understanding Asymptotes in Exponential Functions

An asymptote is a line that a graph approaches but never really touches or crosses. In the case of exponential functions, we often encounter a horizontal asymptote. This is particularly true for functions of the form \( f(x) = ab^x + c \). In our specific function, \( f(x) = -\left(\frac{1}{2}\right)^x + 4 \), the term "\(+4\)" shifts the entire graph upwards by 4 units. This means the graph will level off towards the horizontal line \( y = 4 \) as \( x \to \infty \). Because the coefficient \( a \) is negative, the function approaches the asymptote from below. Remember, the asymptote shows us the behavior of the graph at extremes, but the function will never actually reach \( y=4 \).

Approaching Graph Sketching with Confidence

Graph sketching is a valuable skill in mathematics, helping visualize functions quickly. To sketch the graph of this function, \( f(x) = -\left(\frac{1}{2}\right)^x + 4 \), we're dealing with an exponential decay because of the base \( \frac{1}{2} \) which is less than one. The negative sign indicates a reflection over the x-axis. Start by plotting the key points:

• The y-intercept \((0, 3)\)
• A point such as \((-1, 2)\)
• Another point like \((1, 3.5)\)

These points map the curve's path, helping illustrate its direction. With the function approaching the asymptote \( y = 4 \), sketch a curve starting below this line, indicating the reflected decay. This hands-on method of plotting points and observing trends leads to a clearer graph.

What is a Y-Intercept? Here's a Better Explanation

The y-intercept of a graph is where it crosses the y-axis. This occurs when \( x = 0 \). For our given function, calculating the y-intercept involves substituting 0 into the function, leading to:
\[f(0) = -\left(\frac{1}{2}\right)^0 + 4 = -1 + 4 = 3. \]
Therefore, the y-intercept is the point \((0, 3)\). This point serves as a crucial reference for sketching the graph, giving us one of the starting points to plot the curve. Easy to spot and fundamental to drawing any graph, the y-intercept helps anchor our visualization of the function.