Problem 2 - College Algebra with Modeling and Visualization

Question

How many times does the graph of \(y=a x+b\) with \(a \neq 0\) intersect the \(x\) -axis?

Step-by-Step Solution

Answer

The graph intersects the x-axis once.

Step 1: Understanding the Intersection

When a line intersects the x-axis, it does so at a point where the y-coordinate is zero. Therefore, to find the intersection points of the line and the x-axis, we set the equation of the line equal to zero: \[ y = ax + b = 0 \]

Step 2: Solving for x

With the equation \( ax + b = 0 \), we solve for \( x \) to find where the line intersects the x-axis. Rearrange the terms to solve for \( x \): \[ ax = -b \]\[ x = -\frac{b}{a} \]

Step 3: Analyzing the Solution

Now that we have \( x = -\frac{b}{a} \), this indicates the x-coordinate where the line intersects the x-axis. Since \( a eq 0 \), dividing by \( a \) is valid and the solution is a single real number. Therefore, the line intersects the x-axis at one distinct point.

Key Concepts

Intersection Points

Understanding intersection points is crucial in analyzing the behavior of linear equations. When dealing with the graph of a linear function like \( y = ax + b \), it's important to know where it crosses the axes. The intersection point occurs at a specific location on the graph.
To find an intersection point with the x-axis, we determine where the function's output, \( y \), is zero. This implies that the graph will touch or cross the x-axis where the equation \( ax + b = 0 \) holds.
By solving this equation, we find the x-coordinate of the intersection on the x-axis. On the x-axis, the line crosses only at one point if \( a eq 0 \).
Every line defined by \( y = ax + b \) with \( a eq 0 \) intersects the x-axis at one point, confirming the line's path through the coordinate plane in a predictable manner.

Graphing Linear Functions

Graphing linear functions provides a visual representation of an equation's behavior. Linear equations like \( y = ax + b \) create straight lines, which makes the graphing straightforward.
Each line is defined by its slope, \( a \), and its y-intercept, \( b \). The slope \( a \) indicates how steep the line is, representing the change in \( y \) for a change in \( x \). Meanwhile, \( b \) shows where the line crosses the y-axis.
A useful method for graphing is to first plot the y-intercept \( (0, b) \) on the graph. From this point, use the slope \( a \) to find another point. For example, if \( a = 2 \), move up 2 units for every 1 unit you move to the right.

• Slope \( a \): Rise over run; change in \( y \) over change in \( x \).
• Y-intercept \( b \): Where the line crosses the y-axis.
• Intersection with the x-axis at \( x = -\frac{b}{a} \).

Graphing these lines allows us to visually explore solutions and understand how changes in \( a \) and \( b \) affect the entire function.

Algebraic Solutions

Algebraic solutions involve finding exact answers to equations through mathematical manipulation, primarily using operations such as addition, subtraction, multiplication, and division. In the context of linear functions, algebra helps us identify the relationship between variables and determine specific points of interest.
In the original problem, we work with the equation \( y = ax + b \). To find where this line crosses the x-axis, we set \( y = 0 \) leading to \( ax + b = 0 \).
To solve for \( x \), you would subtract \( b \) from both sides and then divide by \( a \), providing the solution \( x = -\frac{b}{a} \). This precise operation highlights the beauty of algebra in reducing complex problems to simple solutions:

• Set the equation to zero for finding intercepts.
• Isolate the variable \( x \) to solve for it.
• Verify that division is valid, ensuring \( a eq 0 \).

Using algebraic solutions gives us a clear understanding of how a linear equation behaves and helps establish a predictable method for identifying specific points on a graph.