Read, build, and interpret linear functions, including what the slope and the starting value mean in context.
You work with functions of the form f of x equals mx plus b. Questions ask you to evaluate a function at a value, combine two functions, build a function from a description, or interpret what a number in the function means in the situation being described. Interpretation questions are common: given a cost or population model, you decide whether a coefficient is a rate per unit or a fixed starting amount. Some questions hand you a table of values or a graph instead of an equation.
Linear functions are one of the most frequent math topics, typically around 3 to 5 questions out of the 44, since Algebra makes up roughly 35 percent of the section.
Written for this skill, with the answer explained. 227 more are in the app.
The graph of the linear function f is shown in the xy-plane. What is the y-intercept of the graph of y = f(x)?
The cost to rent a bike is $5 plus $3 per hour. After how many hours does the cost equal $26?
Why: The cost after h hours is 5 + 3h, so set 5 + 3h = 26. Subtract 5 from both sides: 3h = 21, so h = 7. Answering 5 reports the flat fee rather than the number of hours. Answering 6 gives 5 + 3(6) = 23 and 8 gives 29, so neither reaches exactly 26.
The graph shows the total cost y, in dollars, of renting a kayak for x hours. What is the best interpretation of the y-intercept of the graph in this context?
Why: The y-intercept is the point where x = 0, which here is (0, 20), a cost of $20 for 0 hours of rental. That is money charged before any time is used, so the reading is A fee of $20 is charged no matter how many hours the kayak is rented. The slope, from (0, 20) to (2, 50), is 30/2 = 15 dollars per hour, so The rental cost increases by $20 for each hour. mistakes the intercept for the rate, and A fee of $15 is charged no matter how many hours the kayak is rented. uses the hourly rate as the fee. The total cost of renting the kayak for 1 hour is $20. is wrong because 1 hour costs 20 + 15 = 35 dollars.
A freight platform descends at a constant rate. It is 34 feet below the loading dock 8 seconds after the descent begins and 76 feet below the dock 20 seconds after the descent begins. How many feet below the dock was the platform when the descent began?
Why: The platform dropped 76 - 34 = 42 feet in 20 - 8 = 12 seconds, so the rate is 3.5 feet per second. Running back 8 seconds from the 34-foot reading gives 34 - 28 = 6 feet. Choice A is the rate, not a depth. Answering 3.5 means you gave the rate of descent in feet per second instead of the starting depth. Answering 17.2 means you divided the 42-foot change by 20 seconds instead of by the 12-second interval. Answering 62 means you added the 28 feet instead of subtracting to run the model back to time zero.
The slope is the amount the output changes for every one-unit increase in the input. In a cost model it is the price per item or per month. In a population model it is the change in population per year.
A linear equation is a statement you solve for a value. A linear function is a rule that turns inputs into outputs, so you can evaluate it at many inputs, graph it, or interpret its parts. The SAT tests both and often mixes them in one question.
Find the slope by dividing the difference in y-values by the difference in x-values. Then substitute one point and the slope into y equals mx plus b and solve for b. Write the final answer in function notation if the question uses it.
Questions on linear functions alone, with explanations on every one, and the app tracks which ones you keep missing.
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