Problem 7 - Precalculus with Calculus Previews

Question

Sketch the graph of the given function \(f\). Find the \(y\) -intercept and the horizontal asymptote of the graph. State whether the function is increasing or decreasing. $$ f(x)=-5+3^{x} $$

Step-by-Step Solution

Answer

The y-intercept is (0, -4), horizontal asymptote is y = -5, and the function is increasing.

Step 1: Identify the Function Components

The function given is \( f(x) = -5 + 3^x \). This is an exponential function with the base greater than 1, which means it will grow exponentially.

Step 2: Find the y-intercept

To find the \( y \)-intercept, evaluate \( f(x) \) at \( x = 0 \). This gives \( f(0) = -5 + 3^0 = -5 + 1 = -4 \). The \( y \)-intercept is the point \( (0, -4) \).

Step 3: Determine the Horizontal Asymptote

For the function \( f(x) = -5 + 3^x \), analyze the behavior as \( x \to -\infty \). Since \( 3^x \to 0 \) in this limit, \( f(x) \to -5 \). The horizontal asymptote is \( y = -5 \).

Step 4: Analyze Increasing or Decreasing Nature

Since \( 3^x \) is an exponential function with a positive base greater than 1, it is an increasing function. Therefore, \( f(x) = -5 + 3^x \) is also increasing as \( x \) increases.

Step 5: Sketch the Graph

Start by plotting the \( y \)-intercept \( (0, -4) \). Draw a curve that approaches \( y = -5 \) from above as \( x \to -\infty \) and rises sharply to the right since \( 3^x \) grows exponentially. The curve remains above the line \( y = -5 \) but never crosses or touches it.

Key Concepts

Graph Sketching

Sketching the graph of an exponential function like \( f(x) = -5 + 3^x \) can offer a visual representation that aids in understanding its behavior. When sketching this graph, consider a few key steps:

• First, identify the basic shape of \( 3^x \), which is an exponential curve increasing as \( x \) increases.
• Next, apply the transformation given by \( -5 \), which shifts the entire graph vertically downward by 5 units.
• Start plotting the points, beginning with the \( y \)-intercept.
• Then, consider the behavior of the function as \( x \) approaches extreme values.

By applying these steps, the resulting graph should show a curve beginning low and moving upwards and to the right, capturing the essence of exponential growth while respecting the effects of transformations. As it approaches infinity, the graph moves sharply upwards, illustrating the rapid increase typical of exponential functions.

Horizontal Asymptote

A horizontal asymptote in a graph represents a line that the function approaches as \( x \) goes towards positive or negative infinity. For exponential functions like \( f(x) = -5 + 3^x \), analyzing the limits can help determine this asymptote.

• For \( f(x) = -5 + 3^x \), notice that as \( x \to -\infty \), \( 3^x \to 0 \).
• This implies \( f(x) \to -5 \), indicating that the horizontal asymptote is \( y = -5 \).
• Thus, the graph will approach but never touch or cross the line \( y = -5 \), no matter how large or small \( x \) becomes.

Understanding the concept of a horizontal asymptote is crucial as it describes the end behavior of the function in this context.

Y-intercept

The \( y \)-intercept is found by evaluating the function at \( x = 0 \). For \( f(x) = -5 + 3^x \), we compute the \( y \)-intercept by setting \( x = 0 \), resulting in \( f(0) = -5 + 3^0 = -4 \). This gives us the \( y \)-intercept at the point \( (0, -4) \).The \( y \)-intercept provides an important anchor point when graphing, as it shows where the function crosses the \( y \)-axis. It helps establish the starting point of the curve and provides a tangible point through which the graph of the function passes. With this information, you begin to visualize how the graph behaves around this point.

Increasing Function

An exponential function with a base greater than 1, like \( 3^x \) in the expression \( f(x) = -5 + 3^x \), is an increasing function. Here's why:

• The expression \( 3^x \) is increasing because its base is greater than 1, meaning as \( x \) increases, \( 3^x \) increases rapidly.
• Adding \( -5 \) to \( 3^x \) simply shifts the entire curve downward but does not alter the increasing nature of the function.
• This means \( f(x) = -5 + 3^x \) consistently increases as \( x \) moves from left to right.

Understanding why \( f(x) \) is increasing helps predict how all parts of the function behave over its domain, reinforcing the expectation that this function grows without bound as \( x \) increases. This concept is foundational when analyzing even more complex functions.