Problem 25 - Algebra Form and Function

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.

Key Concepts

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.