On the test
Advanced Math is about 35% of the Math section, around 6 to 7 questions in each 22-question module. Math modules mix all four domains and go from easier to harder, so these questions appear all through the module, with the hardest ones near the end.
13 to 15 of the 44 questions (about 6 or 7 in each module). All through the module, easier first and harder at the end.
How I approached it
A small story from Aime
When I first looked at these questions, I asked one thing: how does this thing change? Think about two ways of saving money. In the first, you put 1,000 francs in a tin every week. The tin grows by the same amount each time. That is a straight line. In the second, a friend runs a small trade where the money you hold doubles every month. First month it grows a little, later it grows very fast. That is not a straight line. A football kicked up rises, stops, then falls. The SAT calls these nonlinear functions. When I see one, I first name the kind of change: same amount each time (line), same multiplier each time (exponential), or up then down (quadratic). Then the question becomes much easier.
These questions look scary because of the symbols, but there are only three or four shapes to know. Learn what each shape does, and most of the questions open up for you.
What is inside this part
Functions with an x² term whose graph is a parabola, where you find the vertex, the zeros, or the maximum or minimum.
vertex and minimum or maximum value · zeros and factored form · discriminant: how many real zeros · reading a quadratic from a story such as a ball thrown up
Functions where the variable is in the exponent, so the output is multiplied by the same number each step.
growth and decay · writing the formula from a story · finding a and b from two values · percent change as a multiplier
Putting a number into f(x), or saying what a number in a formula means in a story.
f(a) for a given number · what the starting value and multiplier mean · matching a formula to a table or graph
Cubic and higher polynomial functions, usually given in factored form so you can read the zeros.
zeros of cubic functions · graph shape from the factors · evaluating polynomials
How to know a question is from this part
A question gives a formula, table, graph, or story that is clearly not a straight line, and asks for a value, a maximum or minimum, a zero, or the right formula.
- You see x² or x³ in a function, or a number raised to the power x or t.
- The words maximum height, minimum value, vertex, or zero of the function.
- A story with doubling, tripling, 'each year' growth, or a percent increase or decrease.
- f(x) = ... followed by 'what is f(3)?' or 'which equation defines f?'.
- The question says 'the graph of y = ... in the xy-plane'.
The lesson
1. What is a function, in plain words?
A function is a machine. You put a number in, the machine follows a rule, and one number comes out. We write f(x). The x is the number going in. The f(x) is the number coming out.
For example, if f(x) = 3x + 1, then f(2) means: put 2 in the machine. 3 × 2 + 1 = 7. So f(2) = 7. Nothing more.
On a graph, the input x goes along the bottom and the output f(x) goes up the side. Each point (x, y) on the graph says: when I put in x, I get out y. So y and f(x) mean the same thing.
2. What does nonlinear mean?
A linear function makes a straight line. It changes by the same amount each step, like y = 2x + 1.
A nonlinear function does not make a straight line. The graph bends or curves. On the SAT, three kinds appear most: quadratics (the graph is a U shape called a parabola), exponentials (the graph climbs or falls faster and faster), and a few others such as cubic functions (with x³) and polynomials with several turns.
The first job in every question is to look at the formula and say which kind it is.
3. Quadratics: the U shape
A quadratic has an x² term. The standard form is f(x) = ax² + bx + c. If a is positive, the U opens up and has a lowest point. If a is negative, the U opens down and has a highest point. That highest or lowest point is the vertex.
The x-value of the vertex is x = -b ÷ (2a). Put that x back into the function to get the y-value. That y-value is the minimum (if the U opens up) or the maximum (if it opens down).
There are three forms and each tells you something for free. Standard form ax² + bx + c shows the y-intercept: c. Vertex form a(x - h)² + k shows the vertex: (h, k). Factored form a(x - r)(x - s) shows the zeros, the x-values where the graph touches the x-axis: r and s.
What is the minimum value of f(x) = x² - 4x + 3?
Show the steps
- a = 1 is positive, so the U opens up. There is a minimum.
- Vertex x = -b ÷ (2a) = 4 ÷ 2 = 2.
- Put 2 in: f(2) = 4 - 8 + 3 = -1.
The minimum value is -1 (at x = 2).
4. Zeros and the shape of the parabola
A zero (also called a root or x-intercept) is an x-value where f(x) = 0. To find zeros of a quadratic, set it equal to 0 and factor, or use the quadratic formula x = (-b ± √(b² - 4ac)) ÷ (2a).
The part under the root, b² - 4ac, is called the discriminant. If it is positive, there are 2 zeros. If it is 0, there is exactly 1 zero (the vertex touches the x-axis). If it is negative, there are no real zeros (the graph never touches the x-axis).
The parabola is symmetric. The vertex sits exactly in the middle of the two zeros. So if the zeros are 1 and 3, the vertex is at x = 2.
f(x) = 2(x - 1)(x - 5). Where is the vertex x-value, and what is f at that point?
Show the steps
- The zeros are 1 and 5.
- The vertex is in the middle: (1 + 5) ÷ 2 = 3.
- f(3) = 2(2)(-2) = -8.
The vertex is (3, -8).
5. Exponentials: growth by multiplying
An exponential function looks like f(x) = a · b^x. The a is the starting value (when x = 0). The b is the multiplier each time x goes up by 1.
If b is bigger than 1, it is growth. If b is between 0 and 1, it is decay (it shrinks). A 5% increase each year means b = 1.05. A 20% drop each year means b = 0.80.
Compare with a line: a line adds the same amount each step. An exponential multiplies by the same number each step. If a table of values goes 3, 6, 12, 24, that is multiplying by 2, so it is exponential.
A culture has 200 bacteria and doubles every 3 hours. How many bacteria after 12 hours?
Show the steps
- 12 hours has four 3-hour periods.
- Doubling four times: 200 × 2 × 2 × 2 × 2 = 200 × 16.
- 200 × 16 = 3,200.
3,200 bacteria. As a formula: N = 200 · 2^(t ÷ 3).
6. Building an exponential from two facts
Sometimes the SAT gives two values and asks for a third. Say f(x) = a · b^x, f(1) = 12 and f(3) = 108. Going from x = 1 to x = 3 is two steps, so you multiplied by b two times: b² = 108 ÷ 12 = 9. So b = 3 (b is positive). Then a = 12 ÷ 3 = 4.
Now the full rule is f(x) = 4 · 3^x, and you can find any value.
Using f(x) = 4 · 3^x, find f(4).
Show the steps
- 3^4 = 81.
- 4 × 81 = 324.
f(4) = 324.
7. Other nonlinear functions and function questions
Some questions use cubic functions (like x³) or a polynomial given in factored form such as f(x) = (x - 2)(x + 1)(x - 4). The zeros are still where each bracket becomes 0: here 2, -1 and 4.
Some questions ask you to read a graph or table: pick the function that matches. A fast way is to test one or two points. Put the x-value in each option and see which gives the right y.
Some questions give a word story and ask for a formula or a meaning. In f(x) = 50 · 1.2^x, the 50 is the starting amount and the 1.2 means 20% growth each step. Always say what each number means in the story.
8. Common traps
Trap 1: mixing up the number in front (starting amount) with the multiplier. In 200 · 2^(t ÷ 3), the doubling is every 3 hours, so the exponent is t ÷ 3, not 3t.
Trap 2: forgetting the order of operations. For f(x) = x² - 4x with x = -2, the square is 4 and -4x is +8. Use brackets: (-2)² = 4, not -4.
Trap 3: giving the x-value of the vertex when the question asks for the maximum or minimum value (the y-value). Read the question's last line.
Trap 4: a percent change. A 25% decrease is × 0.75, not × 0.25.
See it
For y = x² - 4x + 3 the zeros are 1 and 3, the vertex is exactly in the middle at x = 2, and the lowest value is -1.
- y = 5 + 3x
- Same amount added each step
- Table: 5, 8, 11, 14
- Graph is a straight line
- y = 5 · 3^x
- Same multiplier each step
- Table: 5, 15, 45, 135
- Graph curves up faster and faster
| Form | Looks like | What it shows you at once |
|---|---|---|
| Standard | ax² + bx + c | y-intercept c; vertex x = -b ÷ (2a) |
| Vertex | a(x - h)² + k | vertex (h, k) |
| Factored | a(x - r)(x - s) | zeros r and s |
- 1Read the last line: what exactly is asked?
- 2Name the type: x² (quadratic), number^x (exponential), other
- 3Pick the form or move that fits (vertex, zeros, multiplier)
- 4Compute carefully with brackets
- 5Check your answer against the question
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