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Linear functions: SAT Math notes
Working with functions like f(x)=mx+b: evaluating them, building them from words, tables and graphs, and explaining what their parts mean.
Study it with worked examples and 8 practice questionsPractise Linear functions questions
What this topic tests
A linear function is a rule that turns an input into an output by multiplying by a fixed number and then adding another: f(x)=mx+b. Each time the input goes up by 1, the output changes by the same amount, m. That steady change is what makes the graph a straight line.
Function notation is just a label. f(4) means the output when the input is 4, so you replace every x with 4. When a question says f(4)=31, it is handing you a point, (4,31), on the graph.
SAT questions on this skill ask you to evaluate a function, build one from a description, find one from points, a table or a graph, or explain what the slope, the intercept, or a statement like f(12)=0 means in a real situation.
Key ideas
- f(a)=c means the point (a,c) is on the graph: input a, output c.
- In f(x)=mx+b, m is the change in output for each 1-unit increase in input, and b=f(0) is the starting value.
- Slope from two points or two rows of a table: m=(f(x2)-f(x1))/(x2-x1).
- If the input changes by some amount, the output changes by m times that amount.
- The x-intercept is where the output is 0. In a story it often means the moment something runs out or stops.
- The phrase 'each additional' usually means the first unit is priced separately, so the rate applies to only x-1 units.
The 4 kinds of Linear functions question
Sorted from every Linear functions question in the site's bank: each kind keeps coming up, and has a method that works every time.
1. Evaluate or work backward with function notation
How to spot it. You see something like f(5), g(4)=31 or a formula such as c(x)=mx+500, and must find an output, an input, or a missing constant. Some versions ask you to read a value off a table or graph.
How to answer it:
- To find an output, replace every x with the given input, in parentheses, and simplify.
- To find an input, set the function equal to the given output and solve for x.
- To find a missing constant, plug in the point you are told about, such as g(4)=31, and solve for the constant.
- If the question then asks about a new input, rewrite the full function with the constant filled in and evaluate it.
- For an unusual input like f(3x), find the x that makes the inside equal the number you want, then use that x on the right side.
The trap. Reversing input and output. In g(4)=31, 4 is the x-value and 31 is the result, not the other way round. With f(3x)=x-6, don't just plug 6 into x-6; first ask which x makes 3x=6.
2. Build a function from a description
How to spot it. A situation with a starting amount and a steady rate of change, like a fee plus a cost per mile, or a height that rises the same amount each second. The question asks which function or equation models it.
How to answer it:
- Find the starting value: the amount when the input is 0. This is b.
- Find the rate: how much the output changes for each 1 unit of input. This is m. Make it negative if the amount is going down.
- Write output =b+mx.
- If the rate kicks in only after the first unit or two, use x-1 or x-2 for the units it applies to, then simplify.
- If you are given two data points instead of a rate, find the slope between them first.
- Test your function with a value you know from the story.
The trap. Using the first-day price as the starting value when the rate is for each additional day. Test x=1: your function must give the first-day price. Also watch rates written backward, like down 1 degree for every 550 feet, which is -1/550 degree per foot, not -550.
3. Find the function from points, a table or a graph
How to spot it. You are given two points, values like f(0)=8 and f(1)=12, a table, or a graph, and asked for the function, its slope, an intercept, or another value on the line.
How to answer it:
- Pick two points and find the slope: change in output divided by change in input.
- Find b. If you know f(0), that is b. If not, plug one point into f(x)=mx+b and solve.
- Write f(x)=mx+b.
- Answer what is asked. For the x-intercept, set f(x)=0. For another point, plug in its x.
- Check with a third point if one is given.
- Desmos shortcut: enter the table and type y1∼mx1+b to get m and b at once.
The trap. Dividing the wrong way: slope is change in y over change in x, not the reverse. Also, the y-intercept is f(0). It is not simply the first value in the table unless that row has x=0.
4. Interpret slope, intercept and function values
How to spot it. A function or graph describes a real situation, and the question asks what a number in it means, what the slope or an intercept means, or what a statement like s(3)=121 says.
How to answer it:
- Write down what the input and output stand for, with units.
- The constant term, which equals f(0), is the output when the input is 0: the starting amount.
- The coefficient of x, the slope, is how much the output changes for each 1 unit of input. If it is negative, it is a decrease.
- A statement like f(a)=c means that when the input is a, the output is c. The x-intercept is the input at which the output reaches 0.
- If the input changes by some amount, the output changes by the slope times that amount.
The trap. Swapping input and output. In s(3)=121, 3 is the minutes and 121 is the spoons, not the reverse. And never call the slope a total; it is always an amount per one unit.
Practise Linear functions
On One for All SAT every question has a step-by-step solution, worked both by hand and in Desmos, the calculator built into the digital SAT, and the notes have worked examples of each kind of question, then practice questions you can check as you go. It's free, and there's nothing to sign up for.
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