Problem 12 - Algebra A Combined Function

Question

Graph each exponential function. $$ y=2^{x}-3 $$

Step-by-Step Solution

Answer

Graph an exponential curve translated 3 units down, with a horizontal asymptote at \(y=-3\).

Step 1: Identify the function type

The given function is an exponential function of the form \(y=a^{x}+c\), where \(a=2\) and \(c=-3\). In this form, \(a\) determines the growth rate, and \(c\) is the vertical shift.

Step 2: Understand the transformation

The function \(y=2^x\) is the basic exponential function. The transformation \(-3\) is a vertical shift which moves the entire graph 3 units down.

Step 3: Create a table of values

Calculate a few points for \(x\) values around zero to understand the function's behavior. For example, when \(x = -1\), \(y = 0.5 - 3 = -2.5\); when \(x = 0\), \(y = 1 - 3 = -2\); when \(x = 1\), \(y = 2 - 3 = -1\); and when \(x = 2\), \(y = 4 - 3 = 1\).

Step 4: Plot the points on the coordinate plane

Locate and mark the points calculated from the table: \((-1, -2.5)\), \((0, -2)\), \((1, -1)\), \((2, 1)\) on the coordinate plane.

Step 5: Draw the graph

Draw a smooth curve through the plotted points, ensuring the curve gets closer to the line \(y=-3\) but never touches it, as \(x\) approaches negative infinity. This line, \(y=-3\), is the horizontal asymptote.

Key Concepts

Graphing Exponential Functions

Exponential functions are prevalent in mathematics due to their rapid change and simplicity. When graphing such a function, like the one in this exercise: \( y=2^x-3 \), the first thing to notice is the exponential growth factor, which is 2 in this case. This factor indicates how fast the function will grow or decay. For every one-unit increase in \( x \), the value of the base, here 2, multiplies to give the next value.To graph an exponential function:

• Identify the basic form of the function, \( y = a^x \).
• Recognize any transformations, such as vertical shifts or reflections.
• Determine a few key points to sketch the curve.

Graphing is easier once you understand the base function before incorporating transformations.

Vertical Shift

A vertical shift in a graph simply means moving the entire graph up or down on the coordinate plane. For the function \( y = 2^x - 3 \), the term \(-3\) indicates a vertical shift downward by 3 units.This shift affects every point on the original graph of \( y = 2^x \). Each \( y \)-coordinate is reduced by 3 units. This adjustment does not change the shape of the graph, only its position. Vertical shifts are vital as they modify the starting point of the graph without altering the growth rate or direction of the curve.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph of a function approaches but never actually reaches. In the given function \( y = 2^x - 3 \), the horizontal asymptote is the line \( y = -3 \).This line represents the value that \( y \) will get arbitrarily close to as \( x \) goes to negative infinity. However, the curve will never intersect the horizontal asymptote unless there's a specific shift that elevates or lowers the entire graph onto that level. Understanding the horizontal asymptote is key to knowing the behavior of exponential functions as \( x \) decreases.

Creating a Table of Values

Creating a table of values is a practical step to understanding any function, including exponentials like \( y = 2^x - 3 \). By substituting different \( x \)-values into the function, you can compute specific corresponding \( y \)-values. This results in a series of coordinates perfect for plotting.For example:

• When \( x = -1 \), calculate \( y = 0.5 - 3 = -2.5 \).
• When \( x = 0 \), calculate \( y = 1 - 3 = -2 \).
• When \( x = 1 \), calculate \( y = 2 - 3 = -1 \).
• When \( x = 2 \), calculate \( y = 4 - 3 = 1 \).

These points are plotted to ensure accuracy and aid in drawing the graph. The method serves as a bridge to visualizing and comprehending the formation and orientation of exponential graphs.