Problem 45 - College Algebra with Modeling and Visualization

Question

Sketch a graph of \(y=f(x)\) $$ f(x)=2^{x} $$

Step-by-Step Solution

Answer

Graph: Exponential function, increasing from \((0, 1)\).

Step 1: Understand the Function

The function given is an exponential function \(f(x) = 2^x\). This means it will grow rapidly as \(x\) increases and will approach zero as \(x\) decreases.

Step 2: Determine Key Characteristics of the Graph

For the function \(f(x) = 2^x\), note that it will never cross the x-axis (since \(2^x > 0\) for all real \(x\)). The graph has a horizontal asymptote at \(y = 0\). Also, the y-intercept is at \(f(0)=1\), as \(2^0 = 1\).

Step 3: Plot Important Points

Identify and plot important points on the graph. Start with the y-intercept (0, 1). Then calculate a few other points: \((-2, 0.25), (-1, 0.5), (1, 2), (2, 4)\) by substituting these \(x\)-values into the function.

Step 4: Draw the Graph

Begin sketching the curve through the points plotted. Since \(f(x) = 2^x\) is an exponential function, the curve will rise sharply as \(x\) increases and flatten out as \(x\) decreases.

Key Concepts

Graphing Exponential Functions

Graphing the exponential function \( f(x) = 2^x \) requires a systematic approach to capture its nature accurately. Begin by identifying the function you are working with. Exponential functions have the form \( f(x) = a^x \), where \( a \) is a constant greater than zero, and \( a eq 1 \). In this case, you are working with 2 as your base.

• Choose several values for \( x \), both positive and negative. Start with easy values like -2, -1, 0, 1, and 2.
• Compute the corresponding \( y \)-values using the function. For example, when \( x = 0 \), \( y = 2^0 = 1 \).

After calculating these points, plot them on a graph. Connect the dots smoothly to reveal the characteristic curve of the exponential function.
As you plot more points, the pattern of the graph will become more evident. The graph will rise sharply as \( x \) increases and approach, but never touch, the x-axis as \( x \) decreases.

Characteristics of Exponential Graphs

The distinct shape of exponential graphs like \( f(x) = 2^x \) comes from specific characteristics:

• Rapid Growth: As \( x \) increases, the value of \( 2^x \) grows exponentially large. This is due to the base being greater than one.
• Y-Intercept: All exponential functions where the base is not modified, cross the y-axis at the point (0,1) because any number to the power of zero is one.
• Unbounded Above: The graph doesn't cap at any top value; it continues to climb as \( x \) increases.

These characteristics help in sketching and understanding exponential functions. Recognizing these traits will make it easier to graph and interpret similar functions in different mathematical contexts.

Asymptotes in Exponential Functions

An essential feature of exponential functions is their horizontal asymptote. For \( f(x) = 2^x \), the horizontal asymptote is at \( y = 0 \). This means the graph approaches the x-axis, getting infinitely close but never actually touching or crossing it. This asymptotic behavior occurs because as \( x \) becomes very large, \( 2^x \) increases without bound, and when \( x \) is negative, \( 2^x \) decreases towards zero. However, it never truly reaches the x-axis. Recognizing this behavior is pivotal when sketching the function, as it defines one boundary of the graph.In summary, the presence of a horizontal asymptote shows that the function approaches but never equals zero. This concept is crucial in understanding not just how exponential graphs look but also how they behave as inputs grow infinitely large or small.