Educerie · SAT · Math
Advanced Math · ADV.3 Nonlinear functions
What you must be able to do
| You must be able to | What it looks like on the test |
|---|---|
| Read a parabola's vertex, zeros and intercept from its form | …which form displays the minimum as a constant? |
| Interpret a quadratic model in context | Maximum height, time to land, break-even |
| Write and read an exponential model | …increases by 6% each year |
| Tell linear from exponential | Same amount each step, or same factor |
| Handle transformations | f(x) + 3, f(x − 2), −f(x), 2f(x) |
1The idea in one paragraph
A nonlinear function is still a machine with an input and an output, and the test asks the same three things it asks about lines: where it starts, how it changes, and where it is highest or lowest. For a parabola that means the vertex; for an exponential it means the multiplier. Everything else in this unit follows from knowing which form of the function shows you which fact.
A quadratic's vertex sits exactly halfway between its zeros. If you can factor it, you can find its maximum without completing anything.
2The three forms, and what each displays
| Form | Written | Displays |
|---|---|---|
| Standard | y = ax² + bx + c | The y-intercept, c |
| Factored | y = a(x − r)(x − s) | The zeros, r and s |
| Vertex | y = a(x − h)² + k | The vertex, (h, k) |
The sign of a says which way it opens: positive opens upwards and has a minimum, negative opens downwards and has a maximum.
The vertex's x-coordinate is −b/2a in standard form, or the midpoint of the zeros in factored form. A question asking for "the maximum value" wants k — the y-coordinate — while one asking "at what time is the height greatest" wants h.
3Quadratics in context
h(t) = −5t² + 20t + 1 gives the height of a ball in metres after t seconds.
Read the units in the question to know which of those three it wants. "How long until it lands" is a zero; "how high does it get" is the vertex's y; "when is it highest" is the vertex's x.
4Exponential functions
y = a · bˣ: a is the starting value, b is the multiplier per step.
| English | b |
|---|---|
| grows 8% a year | 1.08 |
| falls 12% a year | 0.88 |
| doubles each hour | 2 |
| halves every 6 hours | 0.5, with exponent t/6 |
The most common wrong answer writes 0.03 as the multiplier, which would destroy 97% of the population each year.
When the period differs from the unit — halving every 6 hours, with t in hours — the exponent becomes t/6.
5Telling linear from exponential
Look at a table. Equal differences between successive outputs means linear; equal ratios means exponential.
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| f(x) | 5 | 10 | 20 | 40 |
In words: a quantity that grows by a fixed amount is linear, and one that grows by a fixed percentage is exponential. The phrase "per year" appears in both, so read what follows it.
6Transformations
| Written | Does |
|---|---|
| f(x) + k | Moves up k |
| f(x) − k | Moves down k |
| f(x − h) | Moves right h |
| f(x + h) | Moves left h |
| −f(x) | Reflects in the x-axis |
| a·f(x) | Stretches vertically by a |
The horizontal ones run opposite to their sign, which is the whole difficulty. f(x − 3) moves right, not left.
Where points are lost
- Writing 0.06 instead of 1.06 for 6% growth.
- Reporting h when the question wants k, or the reverse.
- Getting the direction of a horizontal shift backwards.
- Assuming linear because the table's inputs are evenly spaced.
- Forgetting that a < 0 means a maximum, not a minimum.
- Using the wrong zero for a landing time — negative times are not answers.
- Missing the /6 when the halving period is not one unit.
Work it right
- Decide the family: is it a parabola or a growth curve?
- For a parabola, find the form you have and the fact you need.
- For an exponential, name a and b before anything else.
- Check the sign of a, and whether the question wants when or how much.
- Graph it to confirm, if the calculator is quicker than the algebra.
Try it
Q1. The function h(t) = −5t² + 30t gives the height, in metres, of a projectile t seconds after launch. What is its maximum height, in metres? (Type your answer.)
Q2. A colony of 900 bacteria grows by 15% every hour. Which function gives the population after t hours?
A) P(t) = 900(0.15)ᵗ B) P(t) = 900(1.15)ᵗ C) P(t) = 900 + 1.15t D) P(t) = 900(15)ᵗ
Q3. The function f is defined by f(x) = (x − 4)(x + 2). What is the x-coordinate of the vertex of its graph?
A) −2 B) 1 C) 2 D) 4
Q4. A sample of 240 g of an isotope halves every 8 days. Which function gives the mass after d days?
A) m(d) = 240(0.5)^(8d) B) m(d) = 240(0.5)^(d/8) C) m(d) = 240(8)^(d/2) D) m(d) = 240 − 0.5d
Q5. The graph of y = g(x) is shifted 3 units right and 2 units down. Which function describes the result?
A) g(x + 3) − 2 B) g(x − 3) − 2 C) g(x − 3) + 2 D) g(x + 3) + 2
In one breath
A parabola's three forms each display one fact — standard gives the y-intercept, factored gives the zeros, vertex form gives the turning point — and the vertex always sits halfway between the zeros. An exponential is a starting value times a multiplier: 6% growth is 1.06, not 0.06, and a half-life of six hours puts t/6 in the exponent. Equal differences in a table mean linear, equal ratios mean exponential, and horizontal shifts always run opposite to their sign.
Answers
Q1. 45. vertex at t = −b/2a, then evaluate
Answering 3 gives the time rather than the height, which is the split this question tests.
Q2. B. 15% growth means a multiplier of 1.15 A would leave 15% of the colony each hour. C is linear growth. D multiplies by 15 every hour.
Q3. B — 1. midpoint of the zeros
A and D are the zeros themselves. C is their sum.
Q4. B. the halving period goes under t
A halves every 1/8 of a day. C grows. D is linear decay, which never reaches a half-life.
Q5. B — g(x − 3) − 2. horizontal shifts run opposite to their sign Right is minus inside; down is minus outside. A shifts left. C and D move up.
Educerie · written from the published College Board* Assessment Framework for the Digital SAT Suite *(Math, Advanced Math, skill/knowledge testing point "Nonlinear functions"). All questions and explanations are original Educerie text. Last reviewed 12 September 2026.
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