The building blocks of every nonlinear question: the exponent rules, function notation and evaluation, factoring by greatest common factor, difference of squares and simple trinomials, solving a quadratic by setting factors to zero, reading a parabola's vertex, intercepts and direction, and recognizing exponential growth and decay.
Advanced Math is worth the same thirteen to fifteen questions as Algebra, and its name scares students more than its content should. "Advanced" on the SAT means nonlinear: quadratics, exponentials, polynomials, radicals and rational expressions. Every one of those rests on a short list of mechanical skills that this lesson teaches: exponent rules, function notation, factoring, and reading a parabola.
If Algebra is about lines, Advanced Math is about curves, and the first thing to learn is that a curve's equation and its graph say the same things in two languages. A quadratic's factors are its x-intercepts; its constant term is its y-intercept; the sign of its leading coefficient is whether it opens up or down. An exponential's base is whether it grows or decays and its coefficient is where it starts. Translating between the equation and the picture is most of the domain.
The Medium lesson adds the quadratic formula, vertex form, exponential models and rational expressions; the Advanced lesson adds parameters, sums and products of roots, and equivalent-expression questions with unknown constants. Both assume you can do everything here quickly.
Exponent rules apply only to the same base
x³ times x⁵ is x⁸ because the bases match; x³ times y⁵ cannot be combined. Adding exponents when multiplying, subtracting when dividing, multiplying when raising a power to a power, and turning a negative exponent into a reciprocal cover almost every exponent question on the test. Write the base and the exponent on separate mental lines: first ask what happens to the base, then what happens to the exponent.
Function notation is a substitution instruction
f(x) = 2x² - 3x + 1 defines a rule; f(4) means "replace every x with 4 and compute". f(a + 1) means replace every x with (a + 1), parentheses included. Reading f(4) as f times 4 is the error the test is fishing for when it offers a choice that equals 4 times f(1). Put the input in parentheses every time you substitute, especially when it is negative.
Factor in a fixed order: GCF, then pattern, then trinomial
Always pull out the greatest common factor first, because it makes what remains smaller and often reveals a pattern. Then look for a difference of squares, a² - b² = (a + b)(a - b). Only then factor a trinomial x² + bx + c by finding two numbers that multiply to c and add to b. Doing the steps out of order is how 2x² - 8 gets factored as (2x - 4)(x + 2) with the 2 still trapped inside.
Zero product: a quadratic equals zero only when a factor does
If (x - 2)(x + 5) = 0, then x - 2 = 0 or x + 5 = 0, so x = 2 or x = -5. This only works when one side is zero. For (x - 2)(x + 5) = 8 you must expand, move the 8 over, and re-factor. Setting factors equal to the nonzero right-hand side is one of the most common Advanced Math mistakes.
Read a parabola in four glances
From y = ax² + bx + c: a positive means it opens up (a minimum vertex), a negative means down (a maximum). c is the y-intercept. The x-intercepts are the solutions of ax² + bx + c = 0, found by factoring. The vertex's x-coordinate is halfway between the x-intercepts, or -b/(2a). Those four facts answer nearly every Foundations-level graph question without plotting a point.
Exponents are repeated multiplication, and the rules follow from that. Multiplying powers with the same base adds the exponents: x^a times x^b = x^(a+b). Dividing subtracts them: x^a / x^b = x^(a-b). Raising a power to a power multiplies them: (x^a)^b = x^(ab). A power of a product distributes to each factor: (xy)^a = x^a y^a. Any nonzero base to the zero power is 1, and a negative exponent means a reciprocal: x^(-a) = 1/x^a. The rules never combine different bases, so 2³ times 3² is just 8 times 9 = 72. Radicals are exponents in another notation: √x is x^(1/2), and the Medium lesson develops rational exponents from there.
A function is a rule that assigns exactly one output to each input. The notation f(x) names the rule f and the input x; f(x) = 2x² - 3x + 1 says the output is found by squaring the input, doubling, subtracting three times the input and adding one. Evaluating means substituting: f(2) = 2(4) - 6 + 1 = 3, and f(-1) = 2(1) + 3 + 1 = 6, where the parentheses around -1 are what keep the signs right. A table of values or a graph defines a function just as well as a formula; on a graph, f(3) is the y-coordinate of the point whose x-coordinate is 3, and solving f(x) = 0 means finding where the graph crosses the x-axis.
Factoring rewrites a sum as a product, and it is the main tool for solving nonlinear equations. First factor out the greatest common factor of every term: 6x² + 9x = 3x(2x + 3). Second, look for a difference of squares, which factors as a² - b² = (a - b)(a + b); so x² - 49 = (x - 7)(x + 7) and 4x² - 25 = (2x - 5)(2x + 5). Third, factor a trinomial of the form x² + bx + c by finding two numbers whose product is c and whose sum is b: for x² + 5x + 6 the numbers are 2 and 3, so it factors as (x + 2)(x + 3); for x² - 5x + 6 they are -2 and -3. The special cases x² + 2ab x + b² = (x + b)² and x² - 2bx + b² = (x - b)² are perfect square trinomials. Check any factoring by multiplying back out with FOIL: first, outer, inner, last.
A quadratic equation, ax² + bx + c = 0, has up to two real solutions. When the quadratic factors, the zero product property finishes it: a product is zero only when a factor is zero, so set each factor equal to zero and solve. For x² - 5x + 6 = 0, factoring gives (x - 2)(x - 3) = 0 and the solutions are x = 2 and x = 3. If the equation is not already equal to zero, move everything to one side first. Quadratics that do not factor nicely are solved with the quadratic formula, in the Medium lesson; at the Foundations level every quadratic factors over the integers.
The graph of y = ax² + bx + c is a parabola, a symmetric U-shaped curve. If a is positive the parabola opens upward and its vertex is its lowest point; if a is negative it opens downward and the vertex is the highest point. The y-intercept is (0, c), since substituting x = 0 leaves only c. The x-intercepts, also called zeros or roots, are the real solutions of ax² + bx + c = 0, so the factored form reveals them at once: y = (x + 1)(x - 3) crosses the x-axis at -1 and 3. The axis of symmetry is the vertical line halfway between the x-intercepts, at x = -b/(2a), and the vertex sits on it; its y-coordinate is found by substituting that x back in. For y = x² - 2x - 3 = (x + 1)(x - 3), the axis is x = 1 and the vertex is (1, -4).
An exponential function has the form f(x) = a times b^x with a nonzero and b positive and not 1. The coefficient a is the starting value, f(0), because b⁰ = 1. The base b is the growth factor: if b is greater than 1 the function grows, multiplying by b for each unit increase in x; if b is between 0 and 1 it decays. A quantity that increases 20 percent per year has b = 1.20; one that loses 20 percent per year has b = 0.80. The difference between linear and exponential is the difference between adding the same amount each step and multiplying by the same factor each step; a table whose outputs have a constant ratio, not a constant difference, is exponential. Exponential graphs never cross the x-axis when a is positive and they pass through (0, a).
The exponent rules, with the reason for each
Every rule is repeated multiplication counted carefully. None of them applies across different bases.
| Rule | Statement | Example | Why |
|---|---|---|---|
| Product | x^a times x^b = x^(a + b) | x³ times x⁵ = x⁸ | 3 factors of x times 5 factors of x is 8 factors |
| Quotient | x^a / x^b = x^(a - b) | x⁷ / x² = x⁵ | Two of the seven factors cancel |
| Power of a power | (x^a)^b = x^(ab) | (x³)⁴ = x¹² | Four groups of three factors |
| Power of a product | (xy)^a = x^a y^a | (2x)³ = 8x³ | Each factor is raised to the power |
| Zero exponent | x⁰ = 1 for x not 0 | 5⁰ = 1 | x^a / x^a = 1 and also x^(a - a) |
| Negative exponent | x^(-a) = 1 / x^a | 2^(-3) = 1/8 | Dividing by more factors than you have |
Factoring patterns in the order to try them
Pull out the greatest common factor first; what remains is smaller and its pattern is easier to see.
| Step | Pattern | Example | Factored |
|---|---|---|---|
| 1. GCF | Common factor of every term | 6x² + 9x | 3x(2x + 3) |
| 2. Difference of squares | a² - b² = (a - b)(a + b) | 4x² - 25 | (2x - 5)(2x + 5) |
| 3. Trinomial | Two numbers with product c and sum b | x² + 5x + 6 | (x + 2)(x + 3) |
| 3. Trinomial, negative middle | Both numbers negative | x² - 7x + 12 | (x - 3)(x - 4) |
| 3. Trinomial, negative last | Numbers of opposite sign | x² + 2x - 15 | (x + 5)(x - 3) |
| Perfect square | a² + 2ab + b² = (a + b)² | x² + 6x + 9 | (x + 3)² |
A parabola and everything its equation says about it
y = x² - 2x - 3 = (x + 1)(x - 3). The leading coefficient 1 is positive, so it opens up. The constant -3 is the y-intercept. The factors give x-intercepts at -1 and 3. The axis of symmetry is halfway between them at x = 1, and the vertex is (1, -4).
Growth by multiplying: doubling every step
Each time x increases by 1, y doubles: 1, 2, 4, 8. The graph passes through (0, 1), the starting value, and never touches the x-axis. A linear function would add the same amount each step; this one multiplies by the same factor.
Linear or exponential? Check differences, then ratios
The first table adds 3 each step, so it is linear with slope 3. The second multiplies by 3 each step, so it is exponential with base 3. Both start at 2, so they are y = 3x + 2 and y = 2 times 3^x.
| x | 0 | 1 | 2 | 3 | Pattern |
|---|---|---|---|---|---|
| Linear y | 2 | 5 | 8 | 11 | constant difference +3 |
| Exponential y | 2 | 6 | 18 | 54 | constant ratio x3 |
Example 1: Simplify with the exponent rules
Which expression is equivalent to (3x²)³ / x⁴ for x not equal to 0?
The 3 is cubed too. Leaving it as 3x⁶ is the standard slip.
Plugging in a small number is a fast check for any equivalent-expression question.
Answer: 27x²
Example 2: Evaluate a function at a negative input
If f(x) = 2x² - 3x + 1, what is f(-2)?
Without parentheses, -2² would be read as -4 and the whole answer would be wrong.
Answer: f(-2) = 15
Example 3: Solve a quadratic by factoring
What are the solutions of x² - 5x = -6?
The zero product property needs a zero. Factoring x² - 5x = x(x - 5) and setting factors to -6 is wrong.
Answer: x = 2 or x = 3
Example 4: Build and use an exponential model
A town's population is 5,000 and grows 4 percent each year. Which function gives the population P(t) after t years, and what is the population after 2 years?
Growth of r percent gives base 1 + r/100; decay of r percent gives 1 - r/100. The base is never 0.04.
Answer: P(t) = 5000(1.04)^t; after 2 years the population is 5,408.
Check your understanding 1
Which expression is equivalent to (2x³)² times x⁴?
Check your understanding 2
The function f is defined by f(x) = x² - 4x + 7. What is the value of f(-3)?
Check your understanding 3
Which of the following is equivalent to 9x² - 16?
Check your understanding 4
What are the solutions of the equation x² + 3x - 10 = 0?
Check your understanding 5
The graph of y = x² - 6x + 8 in the xy-plane crosses the x-axis at which two points?
Check your understanding 6
A car bought for 24,000 dollars loses 15 percent of its value each year. Which function gives its value V, in dollars, t years after purchase?
Combining different bases: 2³ times 3² is 72, not 6⁵.
Forgetting to raise the coefficient to the power: (3x²)³ is 27x⁶, not 3x⁶.
Reading f(4) as f times 4, or squaring a negative input without parentheses so that (-3)² becomes -9.
Factoring a trinomial without pulling out the greatest common factor first, leaving a factor trapped inside.
Setting factors equal to a nonzero number: (x - 2)(x + 5) = 8 does not mean x - 2 = 8. Get zero on one side first.
Mixing up the sign of the x-intercepts: (x + 5) is zero at x = -5.
Writing a 4 percent growth base as 0.04 or 4 instead of 1.04, and a 15 percent decay base as 0.15 instead of 0.85.
Same topic, a different level:
Practise this: drills for this topic in the question bank.