Linear functions

You'll get a linear function as a formula like $f(x)=6x-5$, a table, a straight-line graph, two values like $f(2)=11$, or a story with a one-time amount and an amount per unit.

Every linear function can be written as $f(x)=mx+b$. In the gym's $c(x)=15x+40$, $m$ is 15 and $b$ is 40.

• $m$ is the slopeHow steep a line is, measured by how much it goes up for each unit it moves to the right.. It's how much the outputThe number a function gives back, like the 13 when f(4) = 13. changes each time the inputThe number you put into a function, like the 4 in f(4). goes up by 1. In a story it's the rate, like $15 a month.
• $b$ is the output when the input is 0. Call it the starting value. On a graph, the line crosses the vertical axisOne of the two lines a graph is measured against, one running across and one running up. at $(0,b)$, which is the y-interceptThe output when the input is 0, which is where a graph crosses the vertical axis..

Read the last line first. It might say:

• “What is the value of $f(3)$?”
• “For what value of $x$ does $f(x)=21$?”
• “Which equation defines $f$?”
• “Which of the following is the best interpretation of 45 in this context?”

Some questions ask only for a difference like $f(5)-f(2)$, or only for $b$. Keep the last line in mind so you stop at the number it wants.

In function notationWriting a function as f(x), where the number in the parentheses is the one going in., the number in the parentheses is the input. In $f(4)=19$, 4 goes in and 19 comes out.

When the question gives you the output and asks for the input, set the formula equal to that output and solve. Say $g(x)=\frac{3}{5}x+4$, and the question asks for the value of $x$ where $g(x)=19$.

1. Set the formula equal to the output.

   $$\frac{3}{5}x+4=19$$

2. Subtract 4 from both sides.

   $$\frac{3}{5}x=15$$

3. Multiply both sides by 5.

   $$3x=75$$

4. Divide both sides by 3.

   $$x=25$$

5. Check: 15 plus 4 is 19.

   $$\frac{3}{5}(25)+4=19$$

Sometimes the formula has a second letter you don't know yet, like the $k$ in $f(x)=kx+12$. You still put each input in for $x$. Say the question tells you $f(6)=3f(1)$ and asks for $k$.

1. Put 6 in for $x$.

   $$f(6)=6k+12$$

2. Put 1 in for $x$.

   $$f(1)=k+12$$

3. Write $f(6)=3f(1)$ with those.

   $$6k+12=3(k+12)$$

4. Multiply all of $k+12$ by 3, including the 12.

   $$6k+12=3k+36$$

5. Subtract $3k$ and 12 on both sides.

   $$3k=24$$

6. Divide both sides by 3.

   $$k=8$$

With $k=8$, $f(6)=60$ and $f(1)=20$, and 60 is 3 times 20, so it checks.

Do these by hand. Until you know $k$, there's no line for Desmos to draw.

The two values might come as $f(3)=195$, as table rows, as points on a graph, or as readings in a story. Write each one as a point, input first.

A linear function $f$ models a food bank's supply of canned soup $x$ days after a delivery arrived. If $f(3)=195$ and $f(7)=135$, which equation defines $f$?

1. Write each value as a point.

   $$(3,195) \text{ and } (7,135)$$

2. Slope: the change in output over the change in input. Subtract in the same order on top and bottom.

   $$m=\frac{135-195}{7-3}$$

3. Divide. The supply drops 15 cans a day.

   $$m=\frac{-60}{4}=-15$$

4. Put one point into $f(x)=mx+b$.

   $$195=-15(3)+b$$

5. Multiply.

   $$195=-45+b$$

6. Add 45 to both sides.

   $$b=240$$

7. Write the equation.

   $$f(x)=-15x+240$$

8. Check the point you didn't use. It matches.

   $$-15(7)+240=135$$

If the question gives you the slope and one point, skip to the row where a point goes into $f(x)=mx+b$.

Don't skip the last row. The point you used to find $b$ will always fit, because you made it fit. Only the other point can catch a slip.

You need these two formulas by heart. The reference sheetThe page of formulas you can open at any point in the Math section.Read the lesson doesn't print them.

The slope is the change for one unit of input, so you divide by how far the input moved. At the food bank, the supply went from 195 at day 3 to 135 at day 7.

In a table, look at how far the input moves from row to row. It isn't always 1.

1. The output goes up 12 from row to row.

   $$22-10=12$$

2. The input goes up 4 from row to row.

   $$6-2=4$$

3. Divide the output change by the input change.

   $$m=\frac{12}{4}=3$$

4. Put in the first row.

   $$10=3(2)+b$$

5. Subtract 6 from both sides.

   $$b=4$$

6. Write the equation.

   $$f(x)=3x+4$$

Using 12 as the slope gives $f(x)=12x-14$, which fits the first row and misses the other two. And 10 isn't the starting value, because the first row has $x=2$.

When the inputs are years far apart, count years from the first one. Say a model gives a town 46 thousand people in 1994 and 94 thousand in 2010, and the question asks for its estimate for 2018.

1. 1994 becomes 0. Then 2010 is 16 and 2018 is 24.

   $$x=\text{years since 1994}$$

2. The town grows by 3 thousand people a year.

   $$m=\frac{94-46}{16-0}=\frac{48}{16}=3$$

3. 1994 is now input 0, so $b=46$.

   $$f(x)=3x+46$$

4. Put in 24 for 2018.

   $$f(24)=3(24)+46=118$$

Once you have the slope, put one point into $f(x)=mx+b$ and solve for $b$. At the food bank, the slope is $-15$ and the supply was 195 at day 3.

The same steps work when the readings are decimals. If the answer is a grid-inA question where you type your own answer instead of picking one of four.Read the lesson, the grid-in lesson covers what fits in the box. Here's one to try.

Desmos mostly catches slips here, so use it when a fraction or a negative number makes one likely. Here's how to test the food bank's choices.

The right choice goes through both dots. A line with the wrong sign misses at least one, so this catches sign slips too.

Desmos can also find $m$ and $b$ from the same table.

Desmos for data shows more of this. To read where a graphed line crosses an axis, click the line, then click the gray dot that appears there.

Now try one, and test its choices in Desmos.

For $f(x)=6x-5$, $f(5)=25$ and $f(2)=7$, so $f(5)-f(2)=18$. You can skip finding the two outputs, though. The $-5$ cancels when you subtract, and each step of 1 in the input adds 6 to the output.

A question can also give the change like this:

It says that when the input goes up 3, the output goes down 12. So if $h(2)=20$, then $h(5)=8$. Say the question gives $h(2)=20$ and asks which equation defines $h$.

1. Change in output over change in input.

   $$m=\frac{-12}{3}=-4$$

2. Put in $h(2)=20$.

   $$20=-4(2)+b$$

3. Add 8 to both sides.

   $$b=28$$

4. Write the equation.

   $$h(x)=-4x+28$$

5. Check: 3 more in, 12 less out.

   $$h(5)=-4(5)+28=8$$

In a story, find the amount that comes once and the amount per unit. The amount per unit multiplies the input, and the one-time amount stands alone. A gym charges $40 to join and $15 a month, so after $x$ months you've paid $c(x)=15x+40$.

The words also tell you the sign. When something is used up or given out, the amount left goes down, so its slope is negative. The food bank gives out 15 cans a day.

Some stories charge one price for the first day and another for each day after. A rental costs $60 for the first day and $25 for each day after that.

1. $60 for day 1, then $25 for each of the other $x-1$ days.

   $$60+25(x-1)$$

2. Multiply out $25(x-1)$.

   $$60+25x-25$$

3. Combine 60 and $-25$.

   $$25x+35$$

The first price can cover more than one unit. At $50 for the first 3 hours and $12 for each hour after, the $12 is charged for $x-3$ hours, so the cost is $50+12(x-3)=12x+14$.

Words like “twice” or “half” add one more step. Write the model for one time first, then multiply the whole thing. A cleaning costs $30 to set up plus $5 per room, so one cleaning of $x$ rooms costs $5x+30$.

1. Two cleanings: double all of it.

   $$2(5x+30)$$

2. Multiply both parts by 2.

   $$10x+60$$

If the $30 setup is charged only once, two cleanings cost $10x+30$. Check which parts the story repeats.

$s(w)=45w+120$ gives the dollars in a student's savings account $w$ weeks after she starts a summer job.

Check the input's unit too. If a hose adds 5 gallons a minute and $t$ counts hours, the rate in the model is $5\times60=300$ gallons an hour. A choice that says 300 gallons a minute has the wrong unit. Change a rate to the input's unit before you use it.

A question can also ask what the x-interceptThe value of x where a graph crosses the horizontal axis, so the output there is 0.Read the lesson means. That's where the output is 0. For the food bank's $f(x)=-15x+240$:

1. Set the output to 0.

   $$-15x+240=0$$

2. Add $15x$ to both sides.

   $$240=15x$$

3. Divide both sides by 15.

   $$x=16$$

The food bank runs out of soup 16 days after the delivery arrived.

A question that asks only what a number means has nothing to compute, so leave Desmos closed. Try one where the units differ.

Before each one, decide whether you're putting a number in, working backward, building the equation, or saying what a number means. When you build an equation, check it against a value you didn't use.