Question

A linear function is given. (a) Find the slope and y-intercept of each function. (b) Use the slope and y-intercept to graph each function. (c) What is the average rate of change of each function? (d) Determine whether each function is increasing, decreasing, or constant. $$ p(x)=-x+6 $$

Short answer

Slope: -1, y-intercept: 6, Decreasing, Average rate of change: -1.

Step-by-step solution

Identify the Slope and y-intercept

The function given is in the form of the slope-intercept form, which is \(y = mx + b\). Here, \(m=-1\) and \(b=6\). Therefore, the slope (m) is -1 and the y-intercept (b) is 6.

Graph the Function

To graph the function \(p(x)=-x+6\), start by plotting the y-intercept (0,6) on the graph. Then, use the slope. Since the slope is -1, this means that for every increase of 1 unit in x, p(x) decreases by 1 unit. Plot another point by moving 1 unit to the right (x=1) and 1 unit down (y=5). Draw a line through these points to represent the function.

Calculate the Average Rate of Change

The average rate of change for a linear function is the same as its slope. Therefore, the average rate of change of \(p(x)=-x+6\) is -1.

Determine if the Function is Increasing, Decreasing, or Constant

Since the slope of the function is -1, which is less than 0, this implies that the function \(p(x)=-x+6\) is decreasing.

Key concepts

Slope

The slope of a linear function measures how steep the line is. It is denoted by the letter \(m\) in the slope-intercept form \(y = mx + b\). To find the slope, look for the value of \(m\) in the equation. In the given function \(p(x) = -x + 6\), the slope \(m\) is -1.
This value indicates that for every unit increase in \(x\), the value of \(y\) decreases by 1 unit. Thus, a negative slope means the line is going downwards from left to right.

y-intercept

The y-intercept is the point where the line crosses the y-axis. It is represented by the letter \(b\) in the slope-intercept form \(y = mx + b\). For our function, \(b\) is 6.
This means the line crosses the y-axis at the point (0, 6). The y-intercept gives a clear starting point for graphing the function.
When graphing, plot the y-intercept first, and then use the slope to determine the direction and steepness of the line.

Rate of Change

The rate of change in a linear function is crucial for understanding how the function behaves. For linear functions, the rate of change is constant and is equivalent to the slope \(m\).
In \(p(x) = -x + 6\), the rate of change is -1. This tells us how quickly or slowly the values of \(y\) change as \(x\) changes.
Because the rate of change is negative, it signifies that the function is decreasing. An average rate of change helps identify whether a function is increasing, decreasing, or constant.

Graphing

Graphing a linear function helps visualize its slope and y-intercept. Start by plotting the y-intercept on the graph. In our case, plot (0,6).
Next, use the slope to find another point. Since our slope is -1, move one unit to the right (x=1) and one unit down (y=5), and plot this second point.
Finally, draw a straight line through these points to complete the graph of the function. A negative slope will result in a line that slopes downward from left to right, indicating a decreasing function.