SAT Math: Linear Functions

Linear functions form the backbone of the Algebra domain on the Digital SAT, and this free, complete lesson covers everything the College Board's Linear Functions question category tests: the coordinate plane and quadrants, relations, domain and range, identifying functions and the vertical line test, function notation, average rate of change, the slope formula and types of slope, standard form and intercepts, slope-intercept form, point-slope form, writing an equation from two points, and parallel and perpendicular lines. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback, so you can build fluency across the entire topic in one place. Everything here is free, and you can keep practicing afterward with the full SAT Math Question Bank linked below.

1. The Coordinate Plane and Quadrants

A coordinate plane is formed by two number lines that meet at a right angle: the horizontal x-axis and the vertical y-axis. The point where they cross is called the origin, written (0, 0). These two axes divide the plane into four regions called quadrants, numbered with Roman numerals counterclockwise starting from the upper right.

An ordered pair (x, y) gives the exact location of a point: x tells you how far to move left or right from the origin, and y tells you how far to move up or down. A point is in Quadrant I if both coordinates are positive, Quadrant II if x is negative and y is positive, Quadrant III if both are negative, and Quadrant IV if x is positive and y is negative. A point that lies directly on an axis, such as (0, 4) or (−5, 0), is not considered part of any quadrant.

2. Relations, Domain, and Range

A relation is simply a set of ordered pairs. A relation can be shown as a list of ordered pairs, a table of values, a mapping diagram, or a graph, and all four representations describe the exact same information. The domain of a relation is the set of all its x-coordinates (the inputs), and the range is the set of all its y-coordinates (the outputs). When you list a domain or range, each value is written only once, even if it appears more than once in the relation.

3. Identifying Functions and the Vertical Line Test

A function is a special kind of relation in which every element of the domain is paired with exactly one element of the range. In other words, no x-value can appear more than once with two different y-values. It's perfectly fine for a y-value to repeat, as long as each x-value only ever leads to one output.

If a relation is shown as a graph, you can test whether it's a function using the vertical line test: a graph represents a function if and only if no vertical line intersects it more than once. If you can draw even one vertical line that crosses the graph twice, the relation is not a function, because that x-value is paired with two different y-values.

4. Function Notation and Evaluating Functions

An equation that represents a function can be written using function notation. Instead of writing y = 4x − 5, you can write f(x) = 4x − 5. The expression f(x) is read "f of x" and represents the output of the function for a given input x, it does not mean f multiplied by x. To evaluate a function at a specific value, substitute that value everywhere the input variable appears, and then simplify.

5. Average Rate of Change

The average rate of change is a ratio that describes how much one quantity changes, on average, with respect to a change in another quantity. If x is the independent variable and y is the dependent variable, then:

average rate of change = change in ychange in x

To find it from a table, pick any two rows, subtract the y-values, subtract the corresponding x-values in the same order, and divide.

6. The Slope Formula and Types of Slope

Geometrically, the average rate of change between two points on a line is called the slope of the line. The slope m through two points (x₁, y₁) and (x₂, y₂) is:

m = y₂ − y₁x₂ − x₁ = change in ychange in x = riserun

A line can have one of four types of slope, depending on its direction:

7. Standard Form, Intercepts, and Zeros

The standard form of a linear equation is Ax + By = C, where A, B, and C are integers. From standard form, you can quickly find where the line crosses each axis. The x-intercept is the x-coordinate of the point where the graph crosses the x-axis, found by letting y = 0 and solving for x. The y-intercept is the y-coordinate of the point where the graph crosses the y-axis, found by letting x = 0 and solving for y.

These intercepts connect directly to functions: the values of x for which f(x) = 0 are called the zeros of the function. In other words, the zeros of a function are exactly its x-intercepts.

8. Slope-Intercept Form

The slope-intercept form of the equation of a line is:

y = mx + b

where m is the slope of the line and b is the y-intercept. This form is useful because you can read the slope and y-intercept directly from the equation without doing any extra work, and it's the easiest form to graph from or to plug an x-value into.

9. Point-Slope Form

The point-slope form of the equation of a line is:

y − y₁ = m(x − x₁)

where (x₁, y₁) is the coordinates of a known point on the line and m is the slope. This form is especially useful when you're given a point and a slope but don't yet know the y-intercept. Once you have an equation in point-slope form, you can always simplify it into slope-intercept form by distributing and isolating y.

10. Writing an Equation from Two Points

When you're given two points instead of a point and a slope, combine what you've learned: first find the slope using the slope formula, then use either point to write the equation in point-slope form, and finally simplify to slope-intercept form. From there, you can also convert to standard form, Ax + By = C, by clearing fractions and moving the x-term to the left side.

11. Parallel Lines

Lines in the same plane that never intersect are called parallel lines. The key rule: if two nonvertical lines have the same slope, they are parallel, and if two nonvertical lines are parallel, they have the same slope. Two distinct parallel lines always have different y-intercepts, since if they had the same slope and the same y-intercept, they would actually be the same line rather than two separate parallel lines. As special cases, every vertical line is parallel to every other vertical line, and every horizontal line is parallel to every other horizontal line.

12. Perpendicular Lines

Lines that intersect at a right angle are called perpendicular lines. The key rule: if the product of the slopes of two nonvertical, nonhorizontal lines is −1, then the lines are perpendicular. Equivalently, the slopes are negative reciprocals of each other, flip the fraction and switch the sign. As a special case, every vertical line is perpendicular to every horizontal line.

13. Classifying Pairs of Lines

Given the equations of two lines, you can classify their relationship by comparing slopes. Rewrite both equations in slope-intercept form if needed, then compare: equal slopes with different y-intercepts means parallel, slopes that multiply to −1 means perpendicular, and anything else means neither.

Common Mistakes to Avoid

• Reading f(x) as multiplication. f(x) means "the output of function f when the input is x," not f times x. This trips up students who are new to function notation.
• Confusing domain and range. Domain is always the inputs, the x-values. Range is always the outputs, the y-values. A helpful memory trick: domain and x both come earlier alphabetically than range and y.
• Assuming a repeated y-value disqualifies a function. Only a repeated x-value paired with two different y-values makes a relation fail to be a function. Repeated outputs are completely fine.
• Forgetting order of operations when evaluating h(x) = −2x² + 5 at a negative input. Square the input first, then apply the negative coefficient: h(−2) means −2(−2)², not (−2 × −2)².
• Drawing the vertical line test line horizontally. The test always uses a vertical line. A graph can cross many horizontal lines more than once and still be a function.
• Subtracting coordinates in the wrong order. Whichever point you use for y₂ and x₂, keep it consistent. Mixing up the order, like using y₂ − y₁ over x₁ − x₂, flips the sign of the slope.
• Mixing up zero slope and undefined slope. A horizontal line has zero slope, since the rise is 0. A vertical line has undefined slope, since the run is 0 and you can't divide by zero. It helps to remember: "zero" sounds flat, like a horizontal line.
• Swapping the x-intercept and y-intercept. To find the x-intercept, set y = 0. To find the y-intercept, set x = 0. It's easy to do this backward under time pressure, so double-check which variable you set to zero.
• Reporting an intercept as a point when the question asks for a single number, or vice versa. An x-intercept is often just a number, like 8, but is sometimes expected as a coordinate point, like (8, 0). Read the question carefully to see which form is expected.
• Forgetting that a rate of change can be negative. If a quantity is decreasing, like a candle burning down or a car's fuel level dropping, the average rate of change will be negative, and that's the correct, expected result.
• Reading the point in point-slope form with a sign flip. In y − y₁ = m(x − x₁), if the equation shows y + 2, that means y₁ = −2, not 2. Rewrite addition as subtracting a negative before identifying the point.
• Forgetting to distribute before isolating y. When converting point-slope form to slope-intercept form, the slope must be multiplied through both terms inside the parentheses, not just the x term.
• Mixing up which point to substitute. When writing a point-slope equation from two points, either point works, but plug in the whole ordered pair consistently. Don't take the x-coordinate from one point and the y-coordinate from the other.
• Assuming standard form requires A to be positive without checking the question. Some questions specify A > 0 or that A, B, and C are integers with no common factor. Always check the exact requirements before finalizing your answer.
• Losing a negative sign when clearing fractions to reach standard form. When multiplying every term of an equation by a common denominator, apply it carefully to each term, including any negative signs.
• Forgetting to flip the fraction, not just the sign. The perpendicular slope rule requires both steps: take the reciprocal (flip the fraction) and change the sign. A common error is only changing the sign, which gives the opposite slope, not the perpendicular one.
• Assuming equal slopes always means the same line. Two lines can share a slope and still be different, parallel lines, as long as their y-intercepts differ. Always check both the slope and the y-intercept before concluding the lines are identical.
• Not converting to slope-intercept form before comparing. It's tempting to compare the numbers in standard form directly, but the coefficients in Ax + By = C are not the slope. Always solve for y first.
• Mixing up the parallel and perpendicular special cases. Vertical lines are parallel to other vertical lines and perpendicular to horizontal lines, not the other way around.
• Sign errors when computing a negative reciprocal of a negative slope. The negative reciprocal of a negative number is positive. For example, the negative reciprocal of −4 is 14, not −14.

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