Question

Sketch the graph of the function. $$ y=3^{x} $$

Short answer

The graph of the function \(y = 3^{x}\) starts at the point (0,1) and increases without bound as \(x\) becomes large. It gets closer to the x-axis as \(x\) becomes large and negative but never touches or crosses it, displaying a horizontal asymptote at \(y = 0\).

Step-by-step solution

Identify the base of the Exponential Function

The base of our function is 3, which means that our function will always be positive as an exponential function with a positive base is always above the x-axis .

Identify the Y-Intercept

The y-intercept of our function can be found by evaluating the function at \(x = 0\). For all exponential functions \(b^x\), \(b^0 = 1\), so when \(x = 0\), \(y = 1\). Meaning the y-intercept of our function is at (0,1).

Determine the behavior as \(x\) approaches infinity

As \(x\) gets very large, the function \(3^x\) will also get very large. So, the function increases without bound as \(x\) approaches positive infinity.

Determine the behavior as \(x\) approaches negative infinity

As \(x\) gets very small, the function \(3^x\) approaches 0. This is because any positive number to the power of a large negative number tends to zero. This also shows us that our graph has a horizontal asymptote at \(y = 0\).

Sketch the Graph

Now that we have this information, we can sketch our graph. The graph starts at the y-intercept (0,1) and increases as \(x\) becomes large, getting steeper as it does so. As \(x\) becomes large and negative, the graph approaches the x-axis but never touches or crosses it, hence the y-axis at \(y=0\) is a horizontal asymptote.

Key concepts

Exponential Function Behavior

Understanding the behavior of exponential functions is crucial when graphing them. These functions, represented as \( y = b^x \), where \( b > 0 \), show specific patterns depending on the value of the base \( b \). When the base is greater than one, as in the function \( y = 3^x \), the graph will always be positive, indicating it never crosses the x-axis.

As \( x \) increases, the value of \( y \) grows exponentially, which means it gets larger at an increasing rate. Conversely, as \( x \) decreases, heading towards negative infinity, the value of \( y \) decreases and approaches zero, never actually reaching zero, which is known as a horizontal asymptote. This distinct trait reflects the rapid growth or decay of exponential functions and is visually evident in their steep curves on a graph.

Y-Intercept of Exponential Functions

The y-intercept is a fundamental point in the graph of an exponential function because it marks the starting point of the curve on the y-axis. To identify the y-intercept of any exponential function \( y = b^x \), you can evaluate the function when \( x = 0 \). Since any number raised to the power of zero equals one, the y-intercept will always be at \( (0,1) \).

For the given function \( y = 3^x \), plugging in zero for \( x \) yields \( y = 3^0 = 1 \), so the y-intercept is at the coordinate (0,1). This consistent behavior offers a reliable starting point for graphing exponential functions and serves as a reference that the graph will pass through this point regardless of the base.

Horizontal Asymptotes

A horizontal asymptote is a line that the graph of a function approaches but never touches or crosses. For exponential functions like \( y = 3^x \), the horizontal asymptote is typically the x-axis, or \( y = 0 \).

This boundary line represents the limit of the value of \( y \) as \( x \) tends to negative infinity. In other words, as \( x \) becomes more and more negative, the value of the function gets closer to zero but does not become negative. It's an important concept because it defines the behavior of the graph at the extremes and is an essential characteristic to look for when sketching the function. In the case of \( y = 3^x \), this informs us that no matter how far to the left the graph extends, it will always stay above the x-axis.