Problem 12 - College Algebra

Question

Graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. \(f(x)=5^{x}\)

Step-by-Step Solution

Answer

By plugging various \(x\)-values into the function and plotting them, \(f(x) = 5^x\) is determined to be an increasing exponential function due to the base being greater than 1. Graph starts from a very small \(y\) for negative \(x\), passes through (0, 1) and rises rapidly for positive \(x\). The graph of this function on a graphing utility should match this.

Step 1: Make a table of coordinates

Choose several values for \(x\) (it's usually easiest to choose simple ones, like -1, 0, 1, 2) and substitute each into your function. The values you get out become your \(y\). So for \(x = -1\), \(f(x) = 5^{-1} = 0.2\). For \(x = 0\), \(f(x) = 5^0 = 1\). For \(x = 1\), \(f(x) = 5^1 = 5\). And for \(x = 2\), \(f(x) = 5^2 = 25\).

Step 2: Draw the coordinates on a graph

Take each of your pairs \((x, f(x))\) and plot them on your graph. This will give you points at (-1, 0.2), (0, 1), (1, 5), (2, 25). Draw a curve through them.

Step 3: Checking the graph with a graphing utility

Use a graphing utility to input the function \(5^x\), observe the graph of the function and confirm it matches with the hand-drawn graph. The graph should be increasing, start from a very low point at the left (since 5 to a negative power is less than 1), go through (0, 1) and rise rapidly from there.

Key Concepts

Coordinate Graphing

Graphing with coordinates is an essential skill in mathematics, particularly when dealing with functions like exponentials. It involves plotting points on a coordinated system which consists of two axes – the x-axis (horizontal) and the y-axis (vertical). When you have a function like \(f(x) = 5^x\), you can create a table of coordinates to aid in plotting the graph.

For instance, choosing simple and easy numbers for \(x\), such as -1, 0, 1, and 2, helps in calculating corresponding \(y\) values easily. Once you substitute these values into the function:

• \(x = -1\), \(f(x) = 5^{-1} = 0.2\)
• \(x = 0\), \(f(x) = 5^0 = 1\)
• \(x = 1\), \(f(x) = 5^1 = 5\)
• \(x = 2\), \(f(x) = 5^2 = 25\)

You then obtain pairs of coordinates: (-1, 0.2), (0, 1), (1, 5), and (2, 25). Plotting these points on a graph and drawing a smooth curve through them will visually represent the exponential function. This helps in understanding how exponential growth looks graphically.

Graphing Utilities

Graphing utilities, like graphing calculators or software, are incredibly useful tools for visualizing mathematical functions. These utilities can quickly and accurately plot functions, such as \(f(x) = 5^x\). They recreate a precise representation of the function, which is beneficial for double-checking hand-drawn graphs.

To use a graphing utility, simply input the function and let the technology do the work of scale and plotting. It will depict the exponential rise of the function, showing how it starts at a low point (near zero but never zero for negative \(x\)) and dramatically increases as \(x\) becomes positive.

Using these tools is not only about getting the right graph, but also about developing a deeper understanding of the behavior of functions. Additionally, graphing utilities often offer features to zoom in on specific graph sections, making it easier to examine particular points or trends without needing a perfect hand-drawn plot.

Exponential Growth

Exponential growth is a concept where quantities increase rapidly over time, a pattern exemplified by functions like \(f(x) = 5^x\). This type of growth is characterized by its constant proportional rate. Essentially, as \(x\) increases, the value of \(f(x)\) multiplies by a constant factor each time.

• Starts slow: For negative or small positive \(x\), the function value is small, as in \(5^{-1} = 0.2\).
• Rapid escalation: When \(x\) increases, the output jumps significantly, such as from \(f(0) = 1\) to \(f(2) = 25\).

This pattern indicates why exponential growth is frequently seen in real-world phenomena like population growth, compound interest in finance, or even viral spread of information. By graphing exponential functions, one can more clearly see how quickly and dramatically they grow, helping to fully absorb the impact of exponential growth.