Problem 73 - Intermediate Algebra

Question

Graph \(f(x)=3^{x}\). Then use the graph to estimate the value of \(3^{1.5}\)

Step-by-Step Solution

Answer

Using the graph, the estimated value of \(3^{1.5}\) is approximately 5.2.

Step 1: Set Up the Graph

To graph the function \( f(x) = 3^x \), first note the base is 3, indicating an exponential growth. To sketch the graph, select some key points such as \( x = -1, 0, 1, 2 \) and calculate their respective \( y \)-values: \( f(-1) = 3^{-1} = \frac{1}{3} \), \( f(0) = 3^0 = 1 \), \( f(1) = 3^1 = 3 \), and \( f(2) = 3^2 = 9 \). Plot these points on a coordinate plane.

Step 2: Sketch the Graph

Draw a smooth curve through the points plotted in Step 1, showing the characteristic exponential growth of the function. The graph should be increasing and should not touch the y-axis (which it asymptotically approaches as \( x \to -\infty \)).

Step 3: Locate \( x = 1.5 \) on the Graph

Find the point on the x-axis corresponding to \( x = 1.5 \). This value lies between \( x = 1 \) and \( x = 2 \), so visually identify this point on the horizontal axis.

Step 4: Estimate \( f(1.5) \)

Draw a vertical line from \( x = 1.5 \) until it intersects the graph of \( f(x) = 3^x \). From the point of intersection, draw a horizontal line to the y-axis to estimate the \( y \)-value, which represents \( 3^{1.5} \).

Step 5: Approximate and Validate

Estimate the \( y \)-value from the graph. If drawn accurately, this should be around 5.2, since \( 3^{1.5} \) is calculated to approximately 5.196 using a calculator.

Key Concepts

Exponential Growth

Exponential growth describes a process where the increase of a quantity is proportional to its current value, leading to the growth becoming more rapid over time. In a mathematical context, an exponential function such as \( f(x) = 3^x \) illustrates this concept well. Here, the base number 3, which is greater than 1, indicates that as the value of \( x \) increases, the function grows exponentially. This means the value of \( f(x) \) doubles with each additional unit of \( x \) in a consistent pattern. For example, if you consider \( f(x) \) values for different integers:

• At \( x = 0 \), \( f(x) = 3^0 = 1 \)
• At \( x = 1 \), \( f(x) = 3^1 = 3 \)
• At \( x = 2 \), \( f(x) = 3^2 = 9 \)

These values illustrate the rapid increase, typical of exponential functions, which contrasts with linear functions, where growth per unit is constant rather than proportional to the value.

Coordinate Plane

A coordinate plane is a two-dimensional surface formed by two intersecting lines: the x-axis and the y-axis. These axes are perpendicular, creating four quadrants used for locating points. Each point on this plane is defined by a pair of coordinates \((x, y)\). When graphing functions like \( f(x) = 3^x \), you plot different \( (x, y) \) points to visualize the function's behavior.For instance:

• Plot the point \((0, 1)\) for \( x = 0 \) resulting in \( f(x) = 1 \).
• Plot \((1, 3)\) when \( x = 1 \), where \( f(x) = 3 \).
• Continue this for other select points to construct the exponential curve.

By systematically plotting such points, the graph of the function is plotted on the coordinate plane, allowing for visual comprehension of how exponential growth manifests.

Estimating Values from Graphs

Estimating values from graphs involves identifying the approximate output or \( y \)-value of a function for a given input or \( x \)-value. This skill is highly practical, especially in understanding how changes in \( x \) affect the function's output without precise calculations. To estimate \( 3^{1.5} \) using the graph of \( f(x) = 3^x \):

• Locate \( x = 1.5 \) on the horizontal axis, which falls between points you know, such as \( x = 1 \) and \( x = 2 \).
• Draw a vertical line up to the point where it intersects the graph.
• From this intersection, draw a horizontal line towards the y-axis, reading the corresponding \( y \)-value.
• This process provides the estimated output of \( f(1.5) \), essential for checking results or understanding the behavior of functions at irregular \( x \) values.

Asymptotic Behavior

Asymptotic behavior in functions refers to how they behave as the input values become very large or very small. In the case of exponential functions like \( f(x) = 3^x \), the concept of an asymptote is particularly relevant at \( x \to -\infty \). For example,

• The graph of \( 3^x \) never actually touches or crosses the x-axis. Instead, it approaches it - this demonstrates the function's horizontal asymptote along the x-axis or the line \( y = 0 \).

As \( x \) increases positively, \( 3^x \) grows without bound, showing that exponential growth has no upper boundary. Understanding asymptotic behavior helps in anticipating the limits to which a function may extend, providing insights into its long-term behavior.