ACT Math Functions, Graphs and Coordinate Geometry

Functions and coordinate geometry appear throughout the ACT Math test, from evaluating f(x) at a given input to finding the equation of a circle. These topics fall under two subcategories of Preparing for Higher Math, Functions and Geometry, and together they test how well you read graphs, apply formulas, and move between algebraic and visual forms of the same relationship. ACT gives Functions 17 to 20 percent of the Math test and Geometry another 17 to 20 percent, the same range as Algebra. Together, these three subcategories carry the most weight of the five.

The Math test gives you 45 total questions, 41 of which count toward your score, within a 50-minute time limit, with a calculator allowed on every question. Working fast on routine formula questions leaves room for the harder graph and geometry items. This guide breaks down function notation and transformations, graph reading skills, the distance, midpoint, and slope formulas, and circle and line equations, then points you to our free drill sets for practice.

How the ACT Weights Functions and Coordinate Geometry

Preparing for Higher Math makes up 80 percent of the ACT Math test, and Integrating Essential Skills covers the remaining 20 percent. Preparing for Higher Math splits into five subcategories, and Functions and Geometry are two of them. ACT reports nine scores for the Math section, the overall section score plus eight reporting category scores, so Functions and Geometry each get their own line on your score report.

The table below shows how each subcategory contributes to your Math score. Every percentage comes from ACT's official description of the Math test. For the Statistics and Probability row, see our ACT statistics and probability guide for what you need to know.

The Functions subcategory assesses function definition, notation, representation, and application across linear, radical, piecewise, polynomial, and logarithmic functions, and it requires manipulating and translating functions and finding and applying graph features. The Geometry subcategory requires solving for missing values in triangles, circles, and other figures, including trigonometric ratios and equations of conic sections such as circle equations. ACT's category descriptions name conic section equations under Geometry but do not list the distance, midpoint, or slope formulas by name, so expect them anywhere in the coordinate plane questions across Algebra, Functions, and Geometry.

Function Notation and Transformations

Reading Function Notation

Function notation writes a relationship as f(x) instead of y. The letter f names the function, and x marks the input value. f(x) reads as f of x and represents the output value the function produces for a given input. For example, if f(x) = 2x + 3, find f(4) by replacing x with 4: f(4) = 2(4) + 3 = 11.

The same substitution works when the input is an expression rather than a number. With f(x) = 2x + 3, f(a + 1) becomes 2(a + 1) + 3, which simplifies to 2a + 5. Replace every x in the rule, then simplify once the substitution is complete.

Types of Functions on the ACT

The ACT Functions subcategory names five function types explicitly, and the list is not exhaustive. Questions include but are not limited to these five, and each type shows up in both algebraic and graph form.

• Linear functions, which graph as straight lines
• Radical functions, which contain a square root or other root
• Piecewise functions, which apply different rules over different parts of the domain
• Polynomial functions, which combine terms with whole-number exponents
• Logarithmic functions, which reverse exponential functions

Transforming Functions

A transformation shifts, stretches, or flips a function's graph without changing its basic shape. Changes applied outside the function, to f(x) itself, move the graph vertically. Changes applied inside the parentheses, to x, move the graph horizontally and in the opposite direction from the sign. The table below lists the transformations the ACT uses most.

For example, g(x) = f(x) - 3 shifts every point on f(x) down 3 units, and h(x) = f(x + 2) shifts every point 2 units to the left. The minus 3 sits outside the function, so it moves the graph down. The plus 2 sits inside the parentheses with x, so it moves the graph left rather than right.

Reading and Interpreting Graphs

The ACT often shows you a graph and asks you to identify a feature instead of asking you to calculate one. Recognizing these features fast leads to fewer wasted steps. Read the axes first, check the scale on each one, then find the points where the graph meets an axis.

Look for these features whenever a graph appears in a question. Each one answers a different style of question.

• x-intercepts, where the graph crosses the x-axis and y equals 0
• y-intercept, where the graph crosses the y-axis and x equals 0
• Domain, the full set of x-values the graph covers
• Range, the full set of y-values the graph covers
• Intervals where the graph rises (increasing) or falls (decreasing)
• Maximum and minimum points, the highest and lowest points on the graph

The ACT Functions subcategory tests finding and applying features of a graph directly, so practice spotting these features before you reach for algebra. A question asking where a function equals zero is asking for the x-intercepts. A question asking for the largest value a function reaches is asking for the maximum point.

Distance, Midpoint, and Slope Formulas

Three formulas connect two points on the coordinate plane: the distance formula, the midpoint formula, and the slope formula. Each one starts from the same two points, (x1, y1) and (x2, y2), and produces a different piece of information. Label your two points before you substitute, then keep the order of subtraction the same all the way through.

Distance Formula

The distance formula finds the straight-line distance between two points: d = √((x2 - x1)² + (y2 - y1)²). It comes from the Pythagorean theorem, treating the horizontal and vertical gaps between the points as the two legs of a right triangle. Squaring erases any negative sign, so the order you subtract the coordinates in does not change the result.

For example, find the distance between (1, 2) and (4, 6). Subtract the coordinates: 4 - 1 = 3 and 6 - 2 = 4. Square and add: 3² + 4² = 9 + 16 = 25. The square root of 25 is 5, so the distance is 5 units.

Midpoint Formula

The midpoint formula finds the point exactly halfway between two points: M = ((x1 + x2)/2, (y1 + y2)/2). Average the x-coordinates for the midpoint's x-value, then average the y-coordinates for its y-value. For example, the midpoint between (2, 3) and (8, 7) is ((2+8)/2, (3+7)/2), or (5, 5). Watch the addition here, because the distance and slope formulas subtract the same coordinates.

Slope Formula

The slope formula measures how steep a line runs: m = (y2 - y1) / (x2 - x1). A positive slope rises left to right, a negative slope falls left to right, a slope of zero marks a horizontal line, and an undefined slope marks a vertical line. For example, the slope between (1, 4) and (3, 10) is (10 - 4) / (3 - 1), or 3. Subtract the y-values and the x-values in the same order, or the sign of your answer flips.

Circle and Line Equations in the Coordinate Plane

Standard Circle Equation

A circle centered at point (h, k) with radius r follows the equation (x - h)² + (y - k)² = r². The ACT Geometry subcategory includes equations of conic sections such as circle equations among its tested skills. For example, a circle centered at (2, -1) with a radius of 5 has the equation (x - 2)² + (y + 1)² = 25. The signs flip inside the parentheses, so a center coordinate of -1 shows up as + 1, and the right side holds r squared rather than r.

Line Equations

Lines on the coordinate plane appear in three common forms. Pick the form matching the information the question hands you.

Parallel lines share the same slope. Perpendicular lines have slopes which multiply to -1, so one slope is the negative reciprocal of the other. A line with a slope of 2/3 runs perpendicular to any line with a slope of -3/2. Solve Ax + By = C for y whenever you need the slope of a line written in standard form.

Practice Functions and Coordinate Geometry on 10exam

We built two free drills for the skills in this guide. Our Functions & Graphs drill covers function notation, transformations, and graph reading. Our Coordinate Geometry drill covers distance, midpoint, slope, and circle and line equations. Both drills are free and untimed, need no signup, and give instant explanations after each question.

For full-length, timed Math practice under real test conditions, take one of our free practice tests on the ACT exam page. Review every missed Functions or Geometry question, then return to the matching drill above. See our ACT Math section guide for the full format, calculator rules, and content breakdown across every subcategory.

Sources

• Description of Math Test | ACT
• What's on the ACT Test? Exam Sections & Structure | ACT