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Chapter 2: Problem 22

Graph the function. Find the slope, \(y\) -intercept and \(x\) -intercept, if any exist. \(f(x)=3-x\)

Short Answer

Expert verified
Slope is -1, y-intercept is (0, 3), x-intercept is (3, 0).

Step by step solution

01

Identify the Basic Form

The given function is in the format of a linear equation: \(f(x) = -x + 3\). This corresponds to the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
02

Determine the Slope

In the equation \(f(x) = 3 - x\), rearrange it to \(f(x) = -x + 3\). Here, the coefficient of \(x\) is \(-1\), which represents the slope of the line. Thus, the slope \(m\) is \(-1\).
03

Find the y-intercept

In the expression \(f(x) = -x + 3\), the constant \(3\) is the y-intercept \(b\). This means the line crosses the y-axis at the point \((0, 3)\).
04

Calculate the x-intercept

To find the x-intercept, set \(f(x) = 0\) and solve for \(x\). \(0 = -x + 3\) leads to \(x = 3\). Thus, the x-intercept is at the point \((3, 0)\).
05

Sketch the Graph

Using the slope \(-1\), y-intercept \((0, 3)\), and x-intercept \((3, 0)\), plot these points on a coordinate plane. Draw a line through these points to graph the function. The line should decrease from left to right due to the negative slope.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Slope
The slope of a line is a key concept in graphing linear equations. It measures the steepness and direction of a line. In our exercise, we have the linear equation \(f(x) = -x + 3\). Here, the slope \(m\) is \(-1\). This means for every unit we move horizontally to the right along the x-axis, the line moves one unit down. The negative sign indicates a downward slope.

The slope tells us several important things about the line:
  • A positive slope means the line ascends from the left to the right.
  • A negative slope, as in this case, indicates the line descends as you move right.
  • A slope of zero would mean the line is perfectly horizontal, not increasing or decreasing.
  • An undefined slope, where the line is vertical, means a division by zero in the slope calculation.
Understanding the slope is crucial as it helps predict how the line behaves across the coordinate plane. Here, since the slope is -1, the line slopes downwards at a 45-degree angle relative to the axes.
Identifying the y-intercept
The y-intercept is the point where the line crosses the y-axis. In the equation \(f(x) = -x + 3\), the y-intercept can be found directly by looking at the constant term, which is \(b = 3\). This tells us that the line intersects the y-axis at \((0, 3)\).

Understanding how to find and interpret the y-intercept is essential because:
  • It gives a clear starting point for drawing the line on a graph. You always begin graphing from the y-intercept.
  • In many real-world scenarios, the y-intercept represents a starting value, such as initial cost or quantity before any changes.
Finding the y-intercept allows you to intuitively grasp how the equation behaves at its initial value, even before any change occurs in response to different values of \(x\).
Finding the x-intercept
Finding the x-intercept involves determining where the line crosses the x-axis. For this, we set \(f(x) = 0\) and solve for \(x\). In our example, starting from the equation \(0 = -x + 3\), rearranging gives \(x = 3\). Thus, the x-intercept is at \((3, 0)\).

Figuring out the x-intercept is useful because:
  • It shows the point where the function's output equals zero, a critical concept in analyzing functions.
  • For real-world problems, the x-intercept might indicate a break-even point, zero level, or a threshold.
  • With the x-intercept, you can more accurately sketch the graph alongside the y-intercept.
Together with the y-intercept, the x-intercept provides a full picture of the line's behavior, making it easier to visualize and analyze the function.

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