Problem 34 - Precalculus : Building Concepts and Connections

Question

Sketch the graph of each function. $$f(x)=-2(3)^{x}+1$$

Step-by-Step Solution

Answer

The graph of the function \(f(x)=-2(3)^{x}+1\) has a y-intercept at (0,-1), and it approaches the horizontal asymptote \(y=1\) as \(x\) increases or decreases. The graph goes downwards as \(x\) increases, showing a reflection over the x-axis compared to the basic graph \(3^x\). Some points to help draw the graph are (-1, 1/3) and (1, -5).

Step 1: Identify the y-intercept and the asymptote

The y-intercept occurs when \(x = 0\), so plug in 0 into the function to find the y-value. This gives \(f(0) = -2(3)^{0} + 1 = -2*1 + 1 = -1\), so the y-intercept is at (0, -1). The asymptote of the function is given by the constant term, which is +1 in this case. This is the horizontal line \(y = 1\), and the graph will approach this line as \(x\) goes to positive or negative infinity.

Step 2: Find some key points to start drawing the graph

Pick a few more \(x\) values and compute the corresponding \(y\) values using the function. For example, when \(x = -1\), \(f(-1) = -2(3)^{-1} + 1 = -2/3 + 1 = 1/3\), and when \(x = 1\), \(f(1) = -2(3)^{1} + 1 = -6 +1 = -5. So, additional points on the graph are (-1, 1/3) and (1, -5).

Step 3: Draw the graph

Plot the y-intercept, the other points, and the asymptote on a coordinate plane. Since the coefficient in front of the base 3 is negative, the graph will be a reflection over the x-axis compared to the basic graph of \(3^x\). The graph will go downwards as \(x\) increases, approaching the asymptote. Draw a smooth curve connecting the points and approaching the asymptote on both sides.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions that describe situations where a quantity grows or decays at a rate proportional to its current value. Such functions are characterized by an equation of the form
\[ f(x) = a \cdot b^{x} \]
where a is a constant that represents the initial amount or y-intercept, b is the base or growth factor, and x represents the exponent or power. When b is greater than 1, the function represents growth; when b is between 0 and 1, it represents decay.

In our example, \( f(x)=-2(3)^{x}+1 \) defines an exponential function with a negative coefficient, which indicates that the function will decay and reflect across the x-axis. The base of 3 implies that it will grow at a rate three times its size for each increase in x.

Y-Intercept of a Function

The y-intercept of a function is the point where the graph of the function crosses the y-axis. To find the y-intercept, you set the x value to zero and solve the function for the resulting y.
\[ f(0) = a \cdot b^{0} + k = a + k \]
where a is the initial amount, b is the base, and k is the vertical shift of the graph. For an exponential function, any base raised to the zeroth power is 1, making the calculation of y-intercept straightforward by simply adding k.

In our exercise, evaluating \( f(0) \) gave us the y-intercept (-1), which is essential for plotting the initial point of the graph.

Horizontal Asymptotes

Horizontal asymptotes are horizontal lines that the graph of a function approaches but never touches as x approaches infinity or negative infinity. This asymptotic behavior is common in exponential functions where the function will approach but never reach this horizontal line. For the general exponential function
\[ f(x) = a \cdot b^{x} + k \]
the horizontal asymptote can be identified by the value k because as x becomes very large (either positively or negatively), the term involving x becomes insignificant, and the function approaches k.

In the solved example, the graph approaches the line \( y = 1 \) but does not touch it, regardless of the value of x, indicating the horizontal asymptote at y equals 1. Understanding horizontal asymptotes is important for recognizing the long-term behavior of the function.

Function Transformations

Function transformations include operations that alter the appearance of a graph without changing its basic shape. These can include shifting, stretching, compressing, and reflecting the graph. For exponential functions, common transformations include:

• Vertical shifts, where adding or subtracting a number k will move the graph up or down.
• Reflections over the x-axis, when the coefficient in front of the base is negative.
• Vertical stretches and compressions, caused by multiplying the function by a number greater or smaller than 1, respectively.

In our function \( f(x)=-2(3)^{x}+1 \), we see a reflection over the x-axis and a vertical shift upwards by 1. Recognizing these transformations allows you to predict and graph the altered behavior of the exponential function effectively.