Problem 41 - College Algebra

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: Understanding Intercepts

To find the intercepts of the equation, we need to determine the points where the graph intersects the axes. For the x-intercept, we set \(y = 0\), and for the y-intercept, we set \(x = 0\). The function given is \(f(x) = -x + 2\), which can be written as \(y = -x + 2\).

Step 2: Finding x-Intercept

To find the x-intercept, set \(y = 0\) in the equation \(y = -x + 2\). This implies: \[0 = -x + 2\]Now solve for \(x\):\[-x = -2\]which simplifies to: \[ x = 2 \]Thus, the x-intercept is the point \((2, 0)\).

Step 3: Finding y-Intercept

To find the y-intercept, set \(x = 0\) in the equation \(y = -x + 2\). This implies:\[y = -0 + 2\]which simplifies to:\[y = 2\]Thus, the y-intercept is the point \((0, 2)\).

Step 4: Summary of Intercepts

The intercepts of the equation \(f(x) = -x + 2\) are the points \((2, 0)\) for the x-intercept and \((0, 2)\) for the y-intercept.

Key Concepts

Graphing Linear Equations

Graphing linear equations is a basic but crucial skill in understanding the behavior of linear functions. A linear equation is usually of the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The graph of a linear equation is a straight line. The equation \(f(x) = -x + 2\) is a straightforward example of a linear equation. Its slope \(m\) is \(-1\), and its y-intercept \(b\) is \(2\). To plot this on a graph:

• Start by marking the y-intercept \((0, 2)\) on the y-axis. This is where the line will cross the y-axis.
• Use the slope to find another point. A slope of \(-1\) means that for every step you take to the right (positive x-direction), you step down (negative y-direction) by 1 unit. From \((0, 2)\), go right 1 unit and down 1 unit to reach \((1, 1)\).
• Draw a line through these points, extending it in both directions across the graph.

This simple method gives you a visual representation of the linear function, making it easier to identify key components like x- and y-intercepts.

Finding x-Intercept

The x-intercept is the point where the graph of the equation crosses the x-axis. At this point, the value of \(y\) is zero. To find the x-intercept of an equation, set \(y = 0\) in the equation and solve for \(x\). For \(f(x) = -x + 2\), set the equation to:\[0 = -x + 2\]Solving for \(x\) involves:

• Rearranging the equation to isolate \(x\): \(-x = -2\)
• Dividing both sides by \(-1\) to solve for \(x\): \(x = 2\)

The x-intercept is clearly at \((2, 0)\). This means the graph will cross the x-axis at this point, a crucial aspect for sketching the graph and understanding the function's behavior.

Finding y-Intercept

The y-intercept is where the graph crosses the y-axis, and at this point, the value of \(x\) is zero. To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\). Using the example \(f(x) = -x + 2\), insert \(x = 0\):\[y = -0 + 2\]Thus, simplifying gives:

• \(y = 2\)

The y-intercept is at \((0, 2)\). This point shows where the line touches the y-axis and is fundamental in determining the starting point of your graph. Always start graphing a line by identifying this point and then using the slope to find other points.

Linear Functions

Linear functions are among the simplest types of functions in algebra, characterized by their straight-line graphs. They are typically expressed as \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept. Every linear function creates a graph that's a straight line, showing a constant rate of change.Key features of linear functions include:

• Slope \(m\): This determines the direction and steepness of the line. A positive slope rises, while a negative slope falls.
• Y-Intercept \(b\): This is where the line intersects the y-axis. It is the value of \(y\) when \(x\) is zero.

The equation \(f(x) = -x + 2\) exemplifies a linear function with a slope of \(-1\) and a y-intercept of \(2\). Understanding these components helps in easily sketching and analyzing the graph without needing to plot numerous points. Linear functions are foundational for more complex mathematical concepts, making mastering them essential for progression in math studies.