Problem 41 - College Algebra with Corequisite Support

Question

For the following exercises, find the \(x\) - and \(y\) -intercepts of each equation. $$ f(x)=-x+2 $$

Step-by-Step Solution

Answer

The x-intercept is (2, 0) and the y-intercept is (0, 2).

Step 1: Find the x-intercept

To find the x-intercept, set \( f(x) \) equal to zero and solve for \( x \). The equation is \( 0 = -x + 2 \). Add \( x \) to both sides to get \( x = 2 \). Therefore, the x-intercept is \( (2, 0) \).

Step 2: Find the y-intercept

To find the y-intercept, set \( x = 0 \) and solve for \( f(x) \). Substitute \( x = 0 \) into the equation \( f(x) = -x + 2 \), which gives \( f(0) = 0 + 2 \). Therefore, the y-intercept is \( (0, 2) \).

Key Concepts

Understanding the x-intercept

In simple terms, the x-intercept is the point where a graph of a function or an equation crosses the x-axis. At this point, the y-coordinate is always zero because the location is directly on the x-axis.
To find the x-intercept of a linear equation such as the one given, substitute 0 for the output (normally expressed as \( f(x) \) or \( y \)).

For example, consider the equation \( f(x) = -x + 2 \). Set \( f(x) \) to 0:

• \( 0 = -x + 2 \)
• Solving for \( x \), we add \( x \) to both sides: \( x = 2 \)

Therefore, the x-intercept is at point \( (2, 0) \), meaning the graph will pass through point \( x = 2 \) on the x-axis. This method can be applied to find the x-intercept in any linear equation.

Understanding the y-intercept

The y-intercept is the point where the graph of a function or equation crosses the y-axis. At this intersection, the x-coordinate is always zero, as it lies vertically on the y-axis.
To find the y-intercept in a linear equation like \( f(x) = -x + 2 \), substitute 0 for \( x \) and solve for \( f(x) \). This indicates where the line will cross the y-axis.

By substituting \( x = 0 \) into the equation:

• \( f(0) = -(0) + 2 \)
• This simplifies to \( f(0) = 2 \)

As a result, the y-intercept is at point \( (0, 2) \). This means that at \( x = 0 \), the graph intersects the y-axis at \( y = 2 \). The process can be universally applied to determine the y-intercept of any linear function or equation.

Understanding linear equations

Linear equations are equations of the first degree, meaning they involve the highest power of the variable being 1. They often represent straight lines on a graph. Linear equations can be written in the format \( y = mx + b \), where \( m \) represents the slope, and \( b \) is the y-intercept.
In this general expression:

• \( m \) determines the tilt or angle of the line.
• \( b \) shows where the line intersects the y-axis, which we've discussed earlier.

In our specific example, \( f(x) = -x + 2 \), we can identify:

• The slope \( m = -1 \), indicating the line descends from left to right.
• The y-intercept \( b = 2 \), marking the point on the y-axis where the line crosses.

Understanding and identifying these components helps in graphing the line, predicting its direction, and solving connected mathematical problems. Additionally, finding x- and y-intercepts allows you to sketch the graph quickly, making linear functions easier to visualize.