ACT Intermediate Algebra & Functions: Quadratics, Systems, and f(x)

This topic is where ACT Math shifts from arithmetic fluency to genuine algebraic reasoning. The questions take a bit longer to work through, and they reward knowing a shortcut over grinding through the algebra by hand. Getting fast here is less about speed drills and more about recognizing which method a given question actually calls for.

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Did you know

Quadratic equations are the single most common question type in this topic. Knowing the quadratic formula, factoring, and how a parabola's graph relates to its equation cold is worth more time invested than almost anything else in ACT Math.

What's tested

Four question types show up constantly. Each one below has the quick version, plus a "Learn more" with the deeper strategy.

Quadratic equations

Solving for x when the equation includes an x², by factoring, the quadratic formula, or reading a graph.

Learn more: choosing factoring versus the formula

Try factoring first if the numbers look clean, two numbers that multiply to the constant term and add to the middle coefficient. It's usually faster than the quadratic formula when it works.
The quadratic formula, x = (−b ± √(b² − 4ac)) / 2a, always works, use it when factoring isn't obvious after a few seconds of trying.
A parabola's x-intercepts are its solutions. If a question shows you a graph and asks for the solutions, you can often read them straight off the graph without any algebra at all.
Vertex form, y = a(x − h)² + k, gives you the vertex (h, k) directly. Recognize this form so you don't waste time converting to it when you don't need to.

Systems of equations

Two equations, find where they meet, sometimes with one equation linear and the other quadratic.

Learn more: substitution, elimination, and the linear-quadratic combo

Substitution works well when one equation is already solved for a variable, or can be easily. Plug that expression into the other equation and solve.
Elimination works well when adding or subtracting the two equations directly cancels a variable, look for matching or opposite coefficients before picking a method.
A linear-quadratic system usually has two solutions, the two points where a line crosses a parabola. If you only find one, double-check, you likely dropped a solution somewhere in the algebra.
Verify your solution in both original equations, not just the one you used to solve.

Function notation

Evaluating f(x), composing functions, and reading function transformations.

Learn more: composition and transformations, not a new kind of math

To evaluate f(3), substitute 3 everywhere you see x in the definition. That's the whole operation, regardless of how complicated f looks.
f(g(x)) means: evaluate g first, then plug that result into f. Work from the inside out, always.
Transformations shift or stretch a graph predictably: f(x) + k shifts up, f(x + k) shifts left, −f(x) flips vertically. Memorize these four, they come up as graph-reading shortcuts constantly.
"f(x) = 12" questions run backward: set the whole expression equal to 12 and solve for x, instead of evaluating forward like you normally would.

Exponential growth and decay

Modeling a quantity that multiplies by a fixed rate each time period, compound interest and population growth being the classic examples.

Learn more: building the model without memorizing a formula

The general shape is: final = initial × (rate)^(time). Growth uses a rate above 1 (like 1.05 for 5% growth), decay uses a rate below 1 (like 0.95 for 5% decay).
Read the problem for the starting value, the rate, and what unit of time each step represents, before writing any equation. Mixing up whether the rate applies per year or per month is the most common setup error.
If the question gives you two data points instead of a stated rate, you can often find what you need by setting up a ratio between them rather than solving for the rate explicitly.

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Gotcha

Here's the classic trap, almost word for word how it shows up: "If x² − 5x + 6 = 0, what is one possible value of x?" Factoring gives (x − 2)(x − 3) = 0, so x = 2 or x = 3, and both numbers are usually sitting right there as answer choices along with a couple of distractors. The question only asks for "one possible value," so either is correct, but students sometimes second-guess themselves into picking a wrong answer because they expect a single unique solution. Quadratics generally have two solutions, and a question asking for "one possible value" is signaling exactly that.

A quick sanity check before you lock in an answer

For quadratics, plug your solution back into the original equation rather than trusting your factoring. For function questions, re-read exactly what's being asked, evaluating forward and solving backward look similar but require opposite steps, and mixing them up is one of the most common errors in this topic.

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Quick tip

When a system or quadratic question feels like it's taking too long algebraically, check whether the answer choices let you plug in and test instead. On multiple choice, testing four numbers is sometimes faster than solving the equation from scratch.

Reading about the traps is step one.

Ember has original Intermediate Algebra and Functions questions, each with the reasoning explained the same way, including exactly which trap answer it's testing.

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