8th Grade · 8.F.A.3

Linear functions in 8th grade

Linear functions are functions whose graphs are straight lines. In 8th grade, your child learns that every equation of the form $y = mx + b$ makes a straight line. The other half of the topic is spotting functions that are not linear, such as $y = x^2$ .

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

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.

What it means, in plain words

A function is a rule that gives one output for each input. Linear functions change the output by the same amount every time the input goes up by 1. Because the change never varies, the points line up in a straight line on a graph. Every linear function can be written as $y = mx + b$ .

In $y = mx + b$ , the letter $m$ is the slope (also called the rate of change): how much $y$ changes when $x$ goes up by 1. The letter $b$ is the value of $y$ when $x$ is 0, which is where the line crosses the vertical axis. In $y = 3x - 2$ , the slope is 3 and the line crosses at $-2$ . A rule like $y = 5 - x$ is linear too, with $m = -1$ and $b = 5$ .

Linear and nonlinear functions look different in every form. A nonlinear function changes by different amounts at each step, so its graph curves. The area of a square, $A = s^2$ , is an example: sides of 1, 2, 3 and 4 give areas of 1, 4, 9 and 16, which go up by 3, then 5, then 7. A rule where $x$ is squared or sits in the bottom of a fraction is not linear.

Your child should be able to:

Explain why an equation $y = mx + b$ always gives a straight line
Rewrite an equation such as $2y = 6x - 4$ in the form $y = mx + b$
Tell linear and nonlinear functions apart from an equation, a table or a graph
Give an example of a function that is not linear

Example

Rewrite 2y = 6x - 4 in the form y = mx + b. What is the slope, and where does the line cross the vertical axis?

Divide every term by 2 so that $y$ stands alone.
$6x \div 2 = 3x$ and $4 \div 2 = 2$ , so $y = 3x - 2$ .
The slope is 3, and the line crosses the vertical axis at $-2$ .

Example

In a table, x is 0, 1, 2 and 3, and y is 1, 2, 5 and 10. Is this function linear?

Find how much $y$ changes at each step: $2 - 1 = 1$ , $5 - 2 = 3$ and $10 - 5 = 5$ .
The changes are 1, 3 and 5, so they are not the same each time.
The function is not linear, and its graph is a curve. (Its rule is $y = x^2 + 1$ .)

Where children usually get stuck

Your child says a table is not linear because the $y$ -values go down.
"Linear" was learned as "goes up steadily". A line that falls steadily is linear too. It has a negative slope.
What helps. Have your child write the change between each pair of rows. If every change is the same number, even a negative one like $-4$ , the function is linear.
For $y = 7 + 2x$ , your child says the slope is 7.
The slope was memorized as "the first number", which only works when the equation is written in the order $y = mx + b$ .
What helps. Ask which number multiplies $x$ . That number is the slope, wherever it sits in the equation.
Your child checks two points, sees that they can be joined with a ruler and calls the function linear.
Any two points can be joined by a straight line, so two points never show that a function is linear.
What helps. Ask for a third point. For $y = x^2$ , the points $(0, 0)$ , $(1, 1)$ and $(2, 4)$ do not fall on one line.

Try four practice questions

Real 8th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Rewrite the equation in y = mx + b form.
$\displaystyle 2y = 6x - 4$
Loading...
Show the answer and how to solve it
Answer. y = 3x - 2
The equation can be written as y = 3x - 2, so it fits y = mx + b with m = 3 and $b = -2. y = -2 + 3x$ is equivalent and also accepted.
1. 1Look at the equation. The goal is to get y by itself.
2. 2Divide every term by 2.
Common mistake. A common mistake is dividing only 6x by 2 and forgetting to also divide the constant 4 by 2. Every term on both sides of the equation must be divided by 2.
The idea underneath. To show an equation is linear, isolate y. If the result is a number times x plus a number, it is in y = mx + b form.
2On level
Look at the equation below. Is it linear or not linear?
$\displaystyle x + y = 8$
A
linear
B
not linear
Show the answer and how to solve it
The answer is linear because x + y = 8 can be rewritten as y = -x + 8, which fits y = mx + b.
1. 1To decide, check whether the equation can be rewritten in the form y = mx + b.
2. 2Subtract x from both sides to isolate y.
3. 3The result is y = -x + 8, which matches y = mx + b with m = -1 and b = 8, so the equation is linear.
Common mistake. A common mistake is choosing 'not linear' because x and y appear on the same side. Subtracting x from both sides immediately gives y = -x + 8, which is linear.
The idea underneath. Linear is about whether an equation can be rewritten as y = mx + b, not whether it already starts in that form.
3On level
Use 1 term tile, 1 operation tile, and the number tile to build a nonlinear function. The rule must not have the form y = mx + b.
Tap or click a tile to drop it into the highlighted blank, or tap an empty blank first to choose a different one. Tap a filled blank to send its tile back.
y
=
x²
|x|
x
2x
+
-
2
Tap an item to place it in the next empty slot, or drag it to a specific slot. With a keyboard: select an item with Enter or Space, then select a slot to place it. Press Delete or Backspace on a slot to remove an item.
Show the answer and how to solve it
One correct answer is $y = x^{2} + 2$ . The other accepted builds are $y = x^{2} - 2$ , $y = |x| + 2$ , and $y = |x| - 2$ because each includes a nonlinear term.
1. 1A linear function has the form $y = mx + b$ .
2. 2From the tiles, $x$ and $2x$ are linear because they are just constants times $x$ . A linear rule changes by the same amount whenever $x$ increases by 1. For $x=0,1,2$ , $x^{2}$ gives $0,1,4$ : the increases are 1, then 3. For $x=-1,0,1$ , $|x|$ gives $1,0,1$ : the changes are $-1$ , then 1. Neither term has constant changes, so both are nonlinear. Adding or subtracting the number tile does not change those differences.
3. 3Choose a nonlinear term tile, then place an operation tile and the number tile. One correct build is shown below.
4. 4These other completed rules also work because each uses a nonlinear term tile.
Common mistake. A common mistake is choosing $y = x + 2$ or $y = 2x - 2$ because the rule looks different after adding or subtracting 2. Those are still linear because $x$ appears only to the first power, so they still fit $y = mx + b$ .
The idea underneath. What matters is the structure of the $x$ -term: if the rule can be simplified to constant times $x$ plus a constant, it is linear no matter how unusual it looks.

4Stretch

Which answer best explains whether the reasoning below about the equation is correct?

	$\displaystyle 3x + 2y = x + 10$
Maya says	$\displaystyle \text{Not linear because } x \text{ appears on both sides.}$

Maya is correct. If x appears on both sides of an equation, the equation cannot be written in y = mx + b form.

Maya is incorrect because a linear equation must pass through the origin, which means the constant term must equal 0.

Maya is correct. A linear equation must already appear in y = mx + b form before it can be classified as linear.

Maya is incorrect. Collecting the x-terms on one side still leaves an equation that can be rewritten in y = mx + b form, so the equation is linear.

Show the answer and how to solve it

Maya is incorrect. Collecting the x-terms gives 2y = -2x + 10, which can be rewritten in y = mx + b form, so the equation is linear.

1Look at the equation and Maya's claim.
2An equation is linear if it can be written in the form $y = mx + b$ .
3Subtract 3x from both sides, then divide every term by 2 to isolate y.
4This matches $y = mx + b$ with $m = -1$ and $b = 5$ , so the equation is linear. Maya's reasoning is incorrect; variables appearing on both sides do not prevent an equation from being linear because the x-terms can always be collected first.

Common mistake. A common mistake is thinking that if x appears on both sides of an equation, the equation cannot be linear. Variables can always be collected to one side, and the result may still fit $y = mx + b$ form.

The idea underneath. Linearity depends on whether an equation can be rewritten as y = mx + b, not on whether it already looks that way.