Linear Patterns and Rate of Change

A linear pattern occurs when the value of a variable increases (or decreases) by exactly the same amount every time. For example the sequence of numbers 1, 3, 5, 7, 9, … shows a linear pattern because the numbers increase by 2 moving from left to right. Another example is the sequence of numbers 12, 6, 0 –6, –12, … where the numbers decrease by 6 (increase by –6) moving from left to right. In both examples, the growth (the difference between adjacent numbers in the sequence) is constant.

Suppose the cost of a meal is $5. Two meals will cost 2 · $5 = $10, three meals will cost 3 · $5 = $15, etc. The sequence $5, $10, $15, $20, … is a linear pattern since the the total cost increases by a constant $5 as the number of meals increase by 1.

Table 1. Example of linear growth.

The growth of a linear pattern is called linear growth. Such growth demands a constant rate of change. Rate of change means the change in value of the dependent variable per the change in value of the independent variable. A constant rate of change means the rate of change is the same number regardless of where the values are selected. For the cost of a meal case shown in table 1, the rate of change is the increase of $5 per 1 meal. Had we moved from 2 meals to 5 meals the price increases from $10 to $25. That is an increase of $15 for 3 more meals, or $5 per meal. The rate of change is a constant at $5 per meal, or a constant rate of change.

Calculating the Rate of Change

A rate of change involves two variables, one is the independent variable, and the other is the dependent variable. The rate of change is the degree to which the dependent variable changes as the independent variable changes.

The change in value of each variable is obtained between any two data points. The change in value of the dependent variable is divided by the change in value of the independent variable. 

When it comes to functions in equation form, the dependent variable is the function value, which depends on the value of the independent variable.

Try It 3

For the function [latex]f(x)=3x-5[/latex], find the rate of change between:

1. [latex]x=-2[/latex] and [latex]x=0[/latex]

2. [latex]x=3[/latex] and [latex]x=4[/latex]

Is the rate of change linear? Show what this means graphically (use Desmos).

Show Answer

1. Rate of change = 3

2. Rate of change = 3

Yes, the rate of change is linear.

Constant Rate of Change and Slope

If the rate of change of a function is constant, then that function is linear.  Consider the data in table 1. The data shows the cost of buying tickets for a school play.

The rate of change of this data is $8 per ticket and is constant across data points. Consequently, this is a linear function. If we graph these data points as coordinates [latex](x,y) = \left ( \text{No. of tickets, Cost }\right)[/latex], the graph shows that the data points lie on a line (Figure 1).

Figure 1. Cost of tickets.

The graph of a linear function is a line. The rate of change of this line is $8 per ticket. This means that for every 1 ticket increase the cost increases by $8.

The slope of a line is a measure of the steepness of the line, and since the steepness is the same all along the line, the slope is constant.  The slope of a line is defined as the ratio of [latex]\frac{\text{rise}}{\text{run}}[/latex] where rise is the change in [latex]y[/latex]-values and run is the corresponding change in [latex]x[/latex]-values. If [latex]y=f(x)[/latex], then slope = [latex]\frac{\text{rise}}{\text{run}}=\frac{\text{change in function values}}{\text{change in }x\text{-values}}[/latex]. This is the exact definition of rate of change, so the rate of change of a linear function is the constant slope between any two data points on the line. The letter [latex]m[/latex] is generally used to represent slope.

For a linear function whose graph is a line,

Slope of the line = [latex]m=\frac{\text{rise}}{\text{run}}[/latex]

Rate of change of the function = [latex]\frac{\text{change in value of the function value}}{\text{change in value of independent variable}}=\frac{\text{rise}}{\text{run}}[/latex]

therefore

Slope of the graphed line = Rate of Change of the function

Using the graph in figure 1, the slope of the line is [latex]m=\frac{\text{rise}}{\text{run}}=\frac{48}{6}=8[/latex].

Figure 2. Slope of a line

The units for slope are always [latex]\dfrac{\text{units for the function output}}{\text{units for the function input}}[/latex], the change of output value for each unit of change in input value. For the graph in figure 1, the units are $ per ticket. So the slope = the rate of change of the function = $8 per ticket.

When the slope of a graphed linear function is positive, the line is moving in an uphill direction from left to right across the coordinate plane. This is also called an increasing linear function. Likewise, a decreasing linear function is one whose slope is negative. The graph of a decreasing linear function is a line moving in a downhill direction from left to right across the coordinate plane. When the slope equals zero, there is zero rate of change so the graph is neither increasing nor decreasing; it stays the same. The result is a horizontal line. When a line is vertical, it’s slope is undefined. This is because the run or change in [latex]x[/latex] is zero, and it is impossible to divide by 0. However, we must remember that a vertical line is NOT a function.

Figure 3. Slopes.