Problem 5 - College Algebra with Modeling and Visualization

Question

If \(f(x)=a x+b\) with \(a \neq 0,\) how are the zero of \(f\) and the \(x\) -intercept of the graph of \(f\) related?

Step-by-Step Solution

Answer

The zero of \(f(x)\) is the x-coordinate of the x-intercept.

Step 1: Understanding the Zero of the Function

To find the zero of the function, we need to solve for the value of \(x\) that makes \(f(x) = 0\). We start with the equation \(f(x) = ax + b\).

Step 2: Setting the Function to Zero

Set the function equal to zero: \(ax + b = 0\). This equation will help us find the value of \(x\) that makes the function zero.

Step 3: Solving for x

Rearrange the equation to solve for \(x\): \(ax = -b\). Next, divide both sides by \(a\) to isolate \(x\): \(x = -\frac{b}{a}\).

Step 4: Identifying the x-intercept

The x-intercept of a graph is the point where the graph crosses the x-axis. At this point, the value of \(y\) is zero, which corresponds to \(f(x) = 0\).

Step 5: Relating Zero to x-intercept

Since the zero of the function is \(x = -\frac{b}{a}\) and this corresponds to the point where the graph crosses the x-axis, the zero of \(f(x)\) is the x-coordinate of the x-intercept of the graph.

Key Concepts

Zero of a Function

The zero of a function is the point where the function's output is zero. For a given linear function, such as \( f(x) = ax + b \), finding the zero means identifying the \( x \)-value that results in \( f(x) = 0 \). This involves solving the equation \( ax + b = 0 \) to find \( x \).
In simpler words:

• Set the function equal to zero: \( ax + b = 0 \).
• Rearrange the equation: \( ax = -b \).
• Divide by \( a \): \( x = -\frac{b}{a} \).

The point \( x = -\frac{b}{a} \) represents the zero of the function, highlighting where the graph of the function crosses the x-axis.

x-Intercept

The x-intercept is a particular point on the graph of a function where it crosses the x-axis. At this point, the value of the function is zero, hence the terminology "intercept"—indicating an intersection with the x-axis.
Here's how to determine the x-intercept:

• Recognize that at the x-intercept, \( f(x) = 0 \).
• Find the value of \( x \) that satisfies this condition.

For the linear function \( f(x) = ax + b \), the x-intercept is determined by the same equation \( ax + b = 0 \), leading to the result \( x = -\frac{b}{a} \). This x-value is crucial because it's where the graph of the function makes contact with the x-axis.

Solving Linear Equations

Solving linear equations forms the foundation of understanding how functions work, particularly in finding zeros and intercepts. Linear equations are in the form \( ax + b = 0 \), where \( a eq 0 \). The goal in solving them is to find the value of \( x \) by performing a series of algebraic steps.
To solve the equation \( ax + b = 0 \):

• Firstly, isolate the term containing \( x \) (here \( ax \)) by moving other terms to the opposite side. This involves subtracting \( b \) from both sides, resulting in \( ax = -b \).
• Finally, divide each side by \( a \) to obtain \( x = -\frac{b}{a} \).

This value is not just a solution to the equation but also represents the zero of the function and its x-intercept, demonstrating the link between solving linear equations and understanding function properties.