Linear relationships are a type of function because for every input, their is exactly one output. When we compare and look at linear equations, they may be referred to as functions.

Linear equations are often written in slope-intercept form which is handy because it helps us identify the slope and y-intercepts of these lines as shown below.

\displaystyle y=mx+b

\bm{m}

is the slope

\bm{b}

is the y-intercept

When comparing linear functions, it will be useful to recall this formula.

Examples

Example 1

In which of the following functions is y increasing faster?

Function A:

x	0	1	2
y	3	10	17

Function B:

-5

-4

-3

-2

-1

-5

-4

-3

-2

-1

Worked Solution

Create a strategy

Find the slope of the two and compare them. A higher slope means a faster rate of change in y.

Apply the idea

For Function A, use points (0, 3) and (1, 10) to find the slope:

\displaystyle m	\displaystyle =	\displaystyle \dfrac{y_2-y_1}{x_2-x_1}	Write the formula for slope
	\displaystyle =	\displaystyle \dfrac{10-3}{1-0}	Substitute the coordinates
	\displaystyle =	\displaystyle 7	Evaluate

For Function B, identify two points on the line and use them to find the slope:

-5

-4

-3

-2

-1

-5

-4

-3

-2

-1

\displaystyle m	\displaystyle =	\displaystyle \dfrac{y_2-y_1}{x_2-x_1}	Write the formula for slope
	\displaystyle =	\displaystyle \dfrac{3-(-5)}{0-(-2)}	Substitute the coordinates
	\displaystyle =	\displaystyle 4	Evaluate
	\displaystyle <	\displaystyle 7	Compare slopes

The slope of Function A is higher which means that it has a faster increase in y.

Example 2

Which of the following has the higher y-intercept?

The line with a slope of 4 that crosses the y-axis at (0, 6).

The line given by the equation y=x+4.

Worked Solution

Create a strategy

For an equation of the form y=mx+b, b is the value of the y-intercept.

Apply the idea

In option A, the y-intercept is 6.

In option B, the equation is of the form y=mx+b where b=4. So the y-intercept is 4

Since 6 \gt 4, option A has the higher y-intercept.

Example 3

Mario wants to determine which of two slow-release pain medications is more rapidly absorbed by the body.

For the liquid form, the amount of the medication in the bloodstream is presented in the graph below.

The results for the capsule form are presented in the table below.

\text{Time (mins)}, t	\text{Amount in} \\ \text{blood (mgs)}, A
4	24.6
7	42.3
10	60
13	77.7

At what rate, in mg per minute, is the liquid form absorbed?

Worked Solution

Create a strategy

Chooose any two points that lie on the line, and use the formula for slope.

Apply the idea

We use the points (0,0)and (2,8) that lie on the line.

\displaystyle \text{Rate of liquid absorption}	\displaystyle =	\displaystyle \dfrac{y_2-y_1}{x_2-x_1}	Write the formula for slope
	\displaystyle =	\displaystyle \dfrac{8-0}{2-0}	Substitute the values
	\displaystyle =	\displaystyle 4\text{ mg/min}	Evaluate

At what rate, in mg per minute, is the capsule form absorbed?

Worked Solution

Create a strategy

Choose any two points from the table and use the slope formula.

Apply the idea

We can use the points (4,24.6) and (7,42.3) from the table.

\displaystyle \text{Rate of capsule absorption}	\displaystyle =	\displaystyle \dfrac{y_2-y_1}{x_2-x_1}	Write the formula for slope
	\displaystyle =	\displaystyle \dfrac{42.3-24.6}{7-4}	Substitute the values
	\displaystyle =	\displaystyle 5.9\text{ mg/min}	Evaluate

In which form is the medication absorbed more rapidly?

Worked Solution

Create a strategy

Use the fact that the medication that is more quickly absorbed will have a higher rate.

Apply the idea

5.9\text{ mg/min} \gt 4\text{ mg/min}

Comparing the two rates from part (a) and part (b), the capsule form is more quickly absorbed than the liquid form as it has higher rate of absorption of 5.9\text{ mg/min}.

Idea summary

In comparing linear relationships, we need to compare the features of linear equations. Linear equations are often written in slope-intercept form which is handy because it helps us identify the slope and y-intercepts.

\displaystyle y=mx+b

\bm{m}

is the slope

\bm{b}

is the y-intercept

If we don't have the equation of the line we can find the slope using the formula m=\dfrac{y_2-y_1}{x_2-x_1}

We can find the y-intercept by finding the y-value when x=0.

Outcomes

8.F.A.2

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

8.F.A.3

Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.

6.04 Compare linear relationships

Ideas

Compare linear relationships

Examples

Example 1

Example 2

Example 3

Outcomes

8.F.A.2

8.F.A.3

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