ACT Math Study Guide | Perfectmathsat.com

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This ACT Math Study Guide focuses on explaining how to solve the hardest ACT Math problems. It is a unique guide, currently under development, but it already covers essential advanced topics that students must master for high ACT scores. Below is a complete explanation of each concept with examples exactly as provided.

Asymptotes

You should be able to identify vertical and horizontal asymptotes for rational functions.

Vertical Asymptotes

Vertical asymptotes occur when the denominator becomes zero because division by zero makes the function approach ±∞.

Example:
For
y = 3x² / (x − 2)(x + 4)
The vertical asymptotes are at:

We set each denominator factor to zero:
x − 2 = 0 → x = 2
x + 4 = 0 → x = −4

Horizontal Asymptotes

Compare the degree of the numerator to the degree of the denominator:

Case 1: Degrees are equal
Divide the leading coefficients.

Example:
3x² / x² = 3 → horizontal asymptote: y = 3

Another example:
y = (5x + 7) / (2x − 3)
→ horizontal asymptote: y = 5/2

Case 2: Degree of denominator is greater
Horizontal asymptote is y = 0.

Example:
y = (11x + 7) / (x² − 1)
Highest-order terms: 11x / x² = 1/x → approaches 0
So horizontal asymptote: y = 0

Negative Reciprocal (Perpendicular Lines)

If two lines are perpendicular, their slopes are negative reciprocals.

Examples:

You may need to:

Complex Numbers

Squaring a Complex Number

Example:
(3 + 2i)²
= (3 + 2i)(3 + 2i)
= 9 + 6i + 6i + 4i²
= 9 + 12i − 4
= 5 − 12i

Use the fact: i² = −1

Finding the Other Factor from a Product

If the product of two complex numbers is 5 and one number is (2 + i),
the other must be its conjugate 2 − i.

Converting Expressions Like 2 ± √−25

√−25 = 5i
So the expression becomes:
2 ± 5i

Sum of Angles in a Polygon

Formula:
Sum of interior angles = 180°(n − 2)
Examples:

Each interior angle in a regular polygon:
180°(n − 2) / n
Also =
180° − 360°/n

Logarithms

Examples:
log₁₆x = −3/4
→ x = 16^(−3/4) = 1/8

log₂16 = 4

If logₓ27 = 3 → x = 3

logₐ(a³ / a⁸) = −5

Laws of Logarithms

Example:
log((x⁵y¹¹)/z²)
= 5logx + 11logy − 2logz

Absolute Value

Example:
|3x − 2| = 7
Solutions:
x = 3
x = −5/3

If |3x − 2| = −4 → no solution

Inequality example:
|3x − 2| < 7
→ −5/3 < x < 3

Harder examples include:

Circles and Circle Graphs

To convert a percentage to degrees in a circle graph:
Multiply by 3.6

Equation of a Circle

Example:
Center (11, −7), radius 10:
(x − 11)² + (y + 7)² = 100

Trigonometry

Law of Sines

sinA/a = sinB/b = sinC/c

Coordinate Geometry

Additional ACT Math Topics

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