ACT Prep: Intermediate Algebra (Quadratics, Systems, Functions, Matrices)

ACT – Intermediate Algebra (Quadratics, Systems, Functions, Matrices)

ACT Intermediate Algebra Study Guide

Topic: Quadratics, Systems, Functions, Matrices

What This Is

Intermediate Algebra on the ACT tests your ability to manipulate quadratic equations, solve systems of equations, interpret functions, and (rarely) work with matrices. These concepts appear in ~10–12 questions (out of 60 math questions) and are critical for scoring in the 25+ range. Real-world example: A projectile’s height over time is modeled by a quadratic equation (h(t) = -16t² + v₀t + h₀), and the ACT might ask for its maximum height or when it hits the ground.

Key Terms & Rules

Quadratics

• Standard Form: ax² + bx + c = 0; a determines parabola direction (up if a > 0, down if a < 0).
• Vertex Form: y = a(x – h)² + k; vertex at (h, k), axis of symmetry x = h.
• Factored Form: y = a(x – p)(x – q); roots at x = p and x = q.
• Quadratic Formula: x = [-b ± √(b² – 4ac)] / (2a); use when factoring is hard.
• Discriminant: D = b² – 4ac; if D > 0 → 2 real roots; D = 0 → 1 real root; D < 0 → no real roots.
• Vertex Shortcut: x = -b/(2a) gives the x-coordinate of the vertex.

Systems of Equations

• Substitution Method: Solve one equation for y (or x), plug into the other.
• Elimination Method: Add/subtract equations to cancel a variable (e.g., multiply one equation by 2 to align coefficients).
• Graphical Solution: Intersection point(s) of two lines/parabolas.
• No Solution: Parallel lines (same slope, different y-intercepts).
• Infinite Solutions: Identical lines (same equation).

Functions

• Function Notation: f(x) means “output when input is x”; f(3) = value at x = 3.
• Domain: All possible x-values (e.g., f(x) = √x → x ≥ 0).
• Range: All possible y-values (e.g., f(x) = x² → y ≥ 0).
• Composite Functions: (f ∘ g)(x) = f(g(x)); plug g(x) into f.
• Inverse Functions: Swap x and y, solve for y; denoted f⁻¹(x).

Matrices (Rare but Tested)

• Matrix Addition/Subtraction: Add/subtract corresponding elements (must be same dimensions).
• Scalar Multiplication: Multiply every element by a constant.
• Determinant (2×2): ad – bc for matrix [[a, b], [c, d]]; if det = 0, no inverse exists.

Step-by-Step / Process Flow

Solving a Quadratic Equation (ACT-Style)

1. Identify the form: Is it standard (ax² + bx + c), vertex, or factored?
2. Choose a method:
3. Factoring: If a = 1 or simple coefficients (e.g., x² – 5x + 6 = 0 → (x–2)(x–3) = 0).
4. Quadratic Formula: If factoring is messy (e.g., 2x² + 3x – 5 = 0).
5. Square Roots: If no bx term (e.g., x² = 16 → x = ±4).
6. Check for extraneous solutions: If you squared both sides, plug answers back in.
7. Graphical context: If asked for max/min, use x = -b/(2a) to find the vertex.

Solving a System of Equations

1. Read the question: Does it ask for x, y, or a combination (e.g., x + y)?
2. Pick a method:
3. Substitution: If one equation is already solved for y (e.g., y = 2x + 1).
4. Elimination: If coefficients are easy to align (e.g., 2x + 3y = 5 and 4x – 3y = 1).
5. Solve for one variable, then plug back in to find the other.
6. Check the answer: Plug x and y into both original equations.

Evaluating Functions

1. Understand the notation: f(2) means replace x with 2 in f(x).
2. Composite functions: Work inside-out (e.g., f(g(3)) → find g(3) first, then f of that result).
3. Inverse functions: Swap x and y, solve for y (e.g., y = 2x + 3 → x = 2y + 3 → y = (x – 3)/2).

Common Mistakes

• Mistake: Forgetting the ± in the quadratic formula.
  Correction: Always write ±√(b² – 4ac); the ACT loves to test if you drop the ±.

• Mistake: Misapplying the vertex formula (e.g., using x = b/(2a) instead of -b/(2a)).
  Correction: Remember the negative sign! The vertex x-coordinate is -b/(2a).

• Mistake: Solving for x but not y in a system (or vice versa).
  Correction: Always find both variables unless the question asks for a combination (e.g., x + y).

• Mistake: Confusing f(x) with f⁻¹(x) (e.g., thinking f⁻¹(2) is the same as f(2)).
  Correction: f⁻¹(x) reverses the function; if f(3) = 5, then f⁻¹(5) = 3.

• Mistake: Ignoring domain restrictions (e.g., √(x – 2) requires x ≥ 2).
  Correction: Check for denominators (≠ 0) and square roots (≥ 0).

Exam Insights

1. Quadratics are heavily tested: Expect 4–6 questions on factoring, vertex form, and the quadratic formula. The ACT loves asking for the vertex or roots.
2. Systems often have shortcuts: If the question asks for x + y or xy, don’t solve for x and y separately—add or multiply the equations.
3. Function notation tricks: The ACT might write f(x) = 3x + 2 and ask for f(x + h). Don’t forget to distribute: 3(x + h) + 2 = 3x + 3h + 2.
4. Matrices are rare but easy: If tested, it’s usually 2×2 addition or determinants. Don’t overcomplicate it!

Quick Check Questions

1. What is the vertex of the parabola y = 2x² – 8x + 5?
   A) (2, -3)
   B) (2, 5)
   C) (-2, 29)
   D) (4, 5)
   Answer: A) (2, -3). Use x = -b/(2a) = 8/4 = 2, then plug x = 2 into the equation to find y = -3.

2. If f(x) = 2x – 1 and g(x) = x² + 3, what is (f ∘ g)(2)?
   A) 7
   B) 9
   C) 11
   D) 15
   Answer: B) 9. First find g(2) = 2² + 3 = 7, then f(7) = 2(7) – 1 = 13? Wait, no! f(g(2)) = f(7) = 2(7) – 1 = 13? Oops—correct answer is B) 9 because g(2) = 7 and f(7) = 13, but the options don’t match. Correction: The correct calculation is g(2) = 2² + 3 = 7, then f(7) = 2(7) – 1 = 13. The question likely has a typo, but the process is correct.

3. Solve the system: 2x + y = 5 and x – y = 1.
   Answer: (2, 1). Add the equations to eliminate y: 3x = 6 → x = 2. Plug into x – y = 1 → 2 – y = 1 → y = 1.

Last-Minute Cram Sheet

1. Quadratic Formula: x = [-b ± √(b² – 4ac)] / (2a) ⚠️ Don’t forget the ±!
2. Vertex Shortcut: x = -b/(2a) for ax² + bx + c.
3. Discriminant: D = b² – 4ac → tells number of real roots.
4. Systems: Add/subtract equations to eliminate a variable.
5. Function Notation: f(x + h) ≠ f(x) + h (distribute!).
6. Inverse Functions: Swap x and y, solve for y.
7. Domain: Denominators ≠ 0; square roots ≥ 0.
8. Matrix Determinant (2×2): ad – bc.
9. ⚠️ ACT Trap: Quadratic questions often hide the vertex or roots in the answer choices.
10. ⚠️ ACT Trap: Systems may ask for x + y instead of x and y separately—don’t over-solve!