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Chapter 5: Problem 25

Identify the slope and \(y\) -intercept and graph the function. $$ f(x)=4-x $$

Short Answer

Expert verified
How would you graph this function? Answer: The slope of the function is -1, and the y-intercept is 4. To graph the function, start by plotting the y-intercept at the point (0, 4). Then, use the slope to find the next point: for every unit increase in x, the value of y will decrease by 1 unit. Plot several points using this method, then draw a straight line through them to represent the graph of the function f(x) = 4 - x.

Step by step solution

01

Identify the function as a linear function

The given function is f(x) = 4 - x. It is a linear function because it can be written as y = mx + b, where m is the slope and b is the y-intercept.
02

Find the slope and y-intercept of the function

In order to identify the slope and y-intercept from the given function f(x) = 4 - x, we will rewrite f(x) in the form y = mx + b, with x and y representing the independent and dependent variables, respectively. $$ y = -1x + 4 $$ Now we can see that the function is in the form y = mx + b. The slope (m) is -1 and the y-intercept (b) is 4.
03

Graph the function

Now that we have identified the slope and y-intercept of the function, we can graph it on the coordinate plane. 1. Start by plotting the y-intercept at the point (0, 4). 2. From the y-intercept, use the slope to find the next point. Since the slope is -1, this means that for every unit increase in x (moving one unit to the right), the value of y will decrease by 1 unit (moving one unit down). 3. After plotting several points using the slope, draw a straight line through them. This will represent the graph of the function f(x) = 4 - x. Now you have identified the slope and y-intercept and graphed the linear function f(x) = 4 - x.

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Key Concepts

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

Slope
The slope of a linear function is an important concept that indicates how steep the line is. It tells us how much the dependent variable, usually denoted as \( y \), changes in response to a change in the independent variable, \( x \). In the formula \( y = mx + b \), the slope is represented by \( m \). For the function \( f(x) = -x + 4 \), the slope, \( m \), is \(-1\). This means for every increase of 1 in \( x \), the value of \( y \) decreases by 1.
Understanding slope is crucial because:
  • It helps predict the behavior of the line on a graph.
  • It allows determination of whether a line ascends, descends, or remains constant.
  • A positive slope indicates an upward line, a negative slope a downward one, and a slope of zero a horizontal line.
In this scenario, the slope of \(-1\) indicates a downward incline from left to right on the coordinate plane.
Y-Intercept
The y-intercept of a linear function is the point where the line crosses the y-axis. It's represented by \( b \) in the linear equation \( y = mx + b \). For our function \( f(x) = -x + 4 \), the y-intercept is \( 4 \). This is the point (0, 4) on the graph.
The y-intercept is an important concept because:
  • It shows where the line starts when \( x = 0 \).
  • This initial value plays a significant role in positioning the entire line on the graph.
  • Identifying the y-intercept helps in quickly plotting the start of your line when graphing.
By knowing the y-intercept, one can easily establish the initial position of the line and see how it behaves as \( x \) changes.
Graphing Linear Equations
Graphing linear equations involves plotting points and drawing a line through them based on the slope and y-intercept. With the function \( f(x) = -x + 4 \), we've determined that the slope is \(-1\) and the y-intercept is \( 4 \). This allows us to start graphing:
  • First, plot the y-intercept, which is the point (0, 4) on the y-axis.
  • Next, use the slope to determine other points. With a slope of \(-1\), move one unit right along the x-axis and one unit down along the y-axis for each successive point.
  • Mark these points on the graph and draw a straight line through them to represent the equation.
Graphing demonstrates the relationship between \( x \) and \( y \). It visually shows with this linear function, how the dependent variable \( y \) changes as the independent variable \( x \) increases, an essential aspect of analyzing linear equations.

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Most popular questions from this chapter

A fast-food fish restaurant serves meals consisting of fish, chips, and hush- puppies. \- One fish, one order of chips, and one pair of hushpuppies costs \(\$ 2.27\). \- Two fish, one order of chips, and one pair of hushpuppies costs \(\$ 3.26\). \- One fish, one order of chips, and two pairs of hushpuppies costs \(\$ 2.76\) (a) How much should a meal of two fish, two orders of chips, and one pair of hush-puppies cost? (b) Show that \(2 x+2 y+z=3(x+y+z)-(x+y+2 z)\) is an identity in \(x, y,\) and \(z\). (c) Use the identity in part (b) to solve part (a). (d) Someone says that you do not need the information that "Two fish, one order of chips, and one pair of hush-puppies costs \(\$ 3.26\) " to solve part (a). Is this true?

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