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Chapter 4: Problem 38

Graph each function using the slope and \(y\) -intercept. $$f(x)=-x+5$$

Short Answer

Expert verified
The slope of the given function \(f(x) = -x + 5\) is \(-1\) and the y-intercept is \(5\). Plot the y-intercept at point (0, 5). Using the slope, find another point on the line at (1, 4). Draw a straight line through both points to represent the graph of the function.

Step by step solution

01

Identify the slope and y-intercept

Looking at the function \(f(x) = -x + 5\), we can identify the slope and the y-intercept: - The slope (\(m\)) is the coefficient of the x term, which is \(-1\). - The y-intercept (\(b\)) is the constant term, which is \(5\).
02

Plot the y-intercept

First, plot the y-intercept on the graph. The y-intercept is the point where the line crosses the y-axis. Since the y-intercept is \(5\), plot a point at the coordinate \((0, 5)\) on the y-axis.
03

Use the slope to find another point on the line

Now use the slope to find another point on the line. The slope is \(-1\), which means the rise (vertical change) is \(-1\) and the run (horizontal change) is \(1\). Starting from the y-intercept point (0, 5), move down one unit (because of the negative sign of the slope) and to the right one unit. This will give us a new coordinate \((1, 4)\). Plot this point on the graph.
04

Draw the line

Now that we have two points on the line, we can draw a straight line passing through both points. This line represents the graph of the function \(f(x) = -x + 5\).

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Key Concepts

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

Graphing linear equations
Graphing a linear equation allows you to visually represent the relationship between variables. A linear equation, like the one provided in the exercise, creates a straight line on a graph. The equation typically follows the formula \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
To graph a linear equation, start by identifying these two key components. You'll first plot the y-intercept, as it's the starting point of your graph on the y-axis.
Then use the slope to determine how the line moves across the graph鈥攚hether it increases or decreases.
  • Identify the slope and y-intercept.
  • Plot the y-intercept on the graph.
  • Apply the slope to find another point on the line.
  • Join the points to create the line.
This process helps you transform an equation into a clear picture of its behavior.
Slope-intercept form
The slope-intercept form is a way of expressing a linear equation in two variables, typically written as \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) represents the y-intercept.
This form is particularly useful because it immediately shows two features of the line:
  • The slope \( m \), indicating how steep the line is.
  • The y-intercept \( b \), showing where the line crosses the y-axis.
Using this form, you can quickly determine how the line behaves: whether it slants up or down or remains horizontal. When the slope \( m \) is positive, the line rises as it moves from left to right. If \( m \) is negative, the line falls. A slope of zero means the line is horizontal.
Recognizing these characteristics helps you become proficient in reading and creating graphs.
Y-intercept
The y-intercept is a special feature of a linear equation. It's the point where the line crosses the y-axis, marked by the coordinate \( (0, b) \). In the equation \( y = mx + b \), the y-intercept is specifically the \( b \) value.
Understanding the y-intercept is essential because it's a fixed point the line always passes through on a graph. When drafting the graph, you begin at this point.
For example, in the equation \( f(x) = -x + 5 \), the y-intercept is \( 5 \), which means the line crosses the y-axis at \( (0, 5) \).
  • The y-intercept is easy to locate as it zeroes out the impact of any \( x \) values.
  • It provides a starting point when plotting a graph.
  • The y-intercept affects where the entire line is positioned in the vertical plane.
Grasping this concept ensures you're able to efficiently sketch the initial position of linear graphs.

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Most popular questions from this chapter

Write the slope-intercept form (if possible) of the equation of the line meeting the given conditions. parallel to \(y=0\) containing \(\left(-3,-\frac{5}{2}\right)\)

Write the slope-intercept form (if possible) of the equation of the line meeting the given conditions. perpendicular to \(y=-x-8\) containing \((4,11)\)

Lines \(L_{1}\) and \(L_{2}\) contain the given points. Determine if lines \(L_{1}\) and \(L_{2}\) are parallel, perpendicular, or neither. $$\begin{aligned}&L_{1}:(-3,9),(4,2)\\\&L_{2}:(6,-8),(-10,8)\end{aligned}$$

Write the slope-intercept form of the equation of the line, if possible, given the following information. contains \((-3,-1)\) and \((2,-3)\)

Determine the domain of each relation, and determine whether each relation describes \(y\) as a function of \(x .\) $$y=-\frac{5}{9-3 x}$$
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