On the test
Algebra is about 35% of the Math section, around 6 to 7 questions in each 22-question module. Math modules mix all four domains and go from easier to harder within the module, so Algebra shows up from the first question to the last.
13 to 15 of the 44 questions (about 6 or 7 in each module). All through the module, easier first and harder at the end.
How I approached it
A small story from Aime
When I think about a linear function, I think about a bus fare. Say the bus has a fixed ticket fee, and then you pay a little more for each stop you travel. If I know the fee and the cost per stop, I can tell you the price for any trip, even one nobody has taken yet. Nothing about the pattern changes: each new stop adds the same amount. That steady adding is what makes it linear. A linear function is just a rule like that, written with letters. On the SAT I look for two things in the rule: the part that starts it off, and the part that grows by the same amount each time. Then the questions become easy.
Linear functions look scary because of the f(x) notation, but it is only a name for a machine that takes a number in and gives a number out. Learn the machine and the fear goes away.
What is inside this part
You put a given input in the rule and find the output.
Finding f(5) · Finding the input for a given output
You write f(x) = mx + b from two points or from a story.
Slope from two points · Starting value and rate
You say what the slope or intercept means in the story.
Meaning of m · Meaning of b
You use two linear functions together, for example cost and income.
Break-even · Where two lines meet
How to know a question is from this part
A rule written with f(x) or g(x), or a table or story with a constant rate.
- Notation like f(x), g(x), or h(t)
- The words "linear function" or "constant rate"
- "What does the number mean in this context?"
- Two values of the function given and a third requested
- A story with a starting amount and an amount added each time
The lesson
1. What is a function?
A function is a rule that takes an input and gives exactly one output. We write f(x), and we say "f of x". The x is the input. The whole f(x) is the output.
For example, f(x) = 4x − 3. If the input is 5, then f(5) = 4(5) − 3 = 17. We just replace x with 5 and calculate.
2. What makes a function linear
A function is linear if it can be written as f(x) = mx + b. Here m and b are fixed numbers.
m is the slope, also called the rate of change. It tells you how much the output changes when the input goes up by 1. b is the y-intercept, also called the starting value. It is the output when the input is 0.
The graph of a linear function is a straight line.
3. Meaning in a story
The SAT loves to ask what a number in the function means. In a story, m is the amount that is added each time, and b is the amount you begin with.
Example: h(w) = 12 + 2w gives the height in cm of a plant after w weeks. The 12 is the starting height. The 2 means the plant grows 2 cm each week.
The function h(w) = 12 + 2w gives a plant's height in cm after w weeks. What does 2 mean?
Show the steps
- Look at the number that multiplies w. That is the slope.
- The slope is the change in output for each 1 in input.
- So the height changes by 2 cm for each week.
The plant grows 2 cm each week.
4. Finding the slope from two points
If you know two points (x₁, y₁) and (x₂, y₂), the slope is (y₂ − y₁) ÷ (x₂ − x₁). It is the change up divided by the change across.
Once you have m, find b by putting one point in y = mx + b. Or, if the question just wants another output, keep adding the slope step by step.
A linear function g has g(−1) = 11 and g(3) = −5. What is g(5)?
Show the steps
- Slope = (−5 − 11) ÷ (3 − (−1)) = −16 ÷ 4 = −4.
- From x = 3 to x = 5 is 2 steps, so the output changes by 2 × (−4) = −8.
- g(5) = −5 − 8 = −13.
−13
5. Finding the input from the output
Sometimes you are given the output and must find the input. Write the output equal to the rule and solve. For f(x) = 5x + 2 and f(a) = 32, write 5a + 2 = 32, so a = 6.
6. When two rules meet
If one function is cost and another is money earned, they are equal at one input. Set the two rules equal and solve. That is the break-even point.
Example: C(n) = 250 + 40n and R(n) = 65n. Set 250 + 40n = 65n. Then 250 = 25n and n = 10.
7. Common traps
Trap 1: mixing up f(a) with f(x) = a. In f(a) = 32, the 32 is the output and a is the input.
Trap 2: dividing the slope upside down. It is change in output ÷ change in input.
Trap 3: reading b as the slope. In y = 3x + 8, the slope is 3.
Trap 4: forgetting units when a story is asked. Say what each number means.
See it
| Part | Name | What it means in a story |
|---|---|---|
| m | Slope or rate | Amount added for each 1 step in x |
| b | y-intercept | Starting amount when x = 0 |
| x | Input | The thing you choose, like weeks |
| f(x) | Output | The result, like height |
- 1Write the two points
- 2Slope m = change in y ÷ change in x
- 3Put one point in y = mx + b
- 4Solve for b
- 5Write the rule and answer the question
Slope is rise divided by run. b is where the line meets the y axis.
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