For linear functions, you will need to use your linear equation skills to solve problems the new SAT throws at you. If you need a refresher on linear equations, check out our post on the Heart of Algebra section on the SAT Math and head back here when you’re ready.
In the most basic terms possible, functions relate a set of inputs with a set of outputs. Usually, you will be able to spot a function when you see something like f(x) or g(x) instead of the usual y variable on one side of the equation.
The key thing to remember about functions is that each input can only give you one output. For example, you can’t plug in 3 into a function and get two different answers. However, to different values of inputs can yield the same output.
On a graph, an easy way to test whether or not a curve is a function is to use the vertical line test. If you draw an imaginary vertical line anywhere on the graph and it crosses the curve twice, then it is not a function.
Most linear equations will pass the vertical line test. The only situation that you have to look out for is a line that is perfectly vertical. In that case, it is not a function.
The standard notation for a linear function is f(x) = kx + f(0).
The f(x) is similar to what the y-variable represents in a linear equation. Test makers can also call it the range or the output of the function.
The k represents the slope. Take a moment to recall the slope-intercept form and note the similarities between the two.
The f(0) is the y-intercept, or where the line passes through the y-axis when x=0.
The x-variable can be called the domain or the input. The x-variable is what you can control, and the y is the value that you get based on what you put in.
Don’t be overwhelmed by the verbiage. Just remember that you’re dealing with basically the same thing as a linear equation. The key here is to get used to all the terms that the test makers can throw at you.
New SAT Math: Linear Function Graphs
Many problems you will encounter ask you to find a value of a certain coordinate point on a graph. In this case, f(x) corresponds to the y-axis and the x corresponds to the x-axis.
All you have to do is go to a given spot on the curve see what the other coordinate equals. For example, if you see an ‘f(5)’, that means that you need to find what the y-coordinate is on the curve labeled “f(x)” where x = 5.
New SAT Math: Linear Function Tabular Notation
Sometimes, you will be given a chart of a function and be expected to interpret it. The chart will only give you a few values to work with.
You will easily be able to see whether or not there is a linear relationship between the input and outputs. By comparing a few values, there should be a linear trend going upward or downward.
The key here is to make sure that you keep your variables, inputs, and outputs straight. If you do that, you can handle any function table that comes your way.