Problem 4 - Algebra A Combined Function

Question

Graph each exponential function. $$ y=3^{x}-1 $$

Step-by-Step Solution

Answer

Plot points for selected \(x\) values and connect them smoothly.

Step 1: Understand the Exponential Function

In the given function, \(y = 3^x - 1\), the base is \(3\), and the exponent is \(x\). This is an exponential function, which means it will grow rapidly as \(x\) increases. The \(-1\) indicates a vertical shift downwards by 1 unit.

Step 2: Create a Table of Values

To graph the function, we start by choosing a few values for \(x\) to compute the corresponding \(y\) values. For example, choose \(x = -1, 0, 1, 2\).- If \(x = -1\), \(y = 3^{-1} - 1 = \frac{1}{3} - 1 = -\frac{2}{3}\).- If \(x = 0\), \(y = 3^0 - 1 = 1 - 1 = 0\).- If \(x = 1\), \(y = 3^1 - 1 = 3 - 1 = 2\).- If \(x = 2\), \(y = 3^2 - 1 = 9 - 1 = 8\).These points are \((-1, -\frac{2}{3})\), \((0, 0)\), \((1, 2)\), and \((2, 8)\).

Step 3: Plot the Points on the Graph

Using the table of values, plot the points on a coordinate plane: \((-1, -\frac{2}{3})\), \((0, 0)\), \((1, 2)\), and \((2, 8)\). These points help to form the shape of the graph.

Step 4: Draw the Exponential Curve

Connect the plotted points with a smooth curve to demonstrate the exponential growth pattern. Make sure the graph approaches the horizontal asymptote at \(y = -1\) as \(x\) goes to negative infinity.

Step 5: Analyze the Graph

The graph of \(y = 3^x - 1\) will pass through the points mentioned and will show exponential growth, with the line never crossing below the horizontal asymptote \(y = -1\). The domain is all real numbers, and the range is \(y > -1\).

Key Concepts

Graphing Exponential Functions

When graphing exponential functions, it's important to understand the basic shape and behavior. An exponential function follows the form \(y = a^x + b\), where \(a\) is the base of the exponent, \(x\) is the exponent, and \(b\) represents any vertical shifts in the graph. In our example \(y = 3^x - 1\), the base is \(3\) which means the function will grow quickly as \(x\) increases.
To correctly graph this function, you need to:

• Identify the base, which tells you the direction and steepness of the curve.
• Consider any shifts, like the \