Nonlinear functions

Nonlinear-function questions test whether you can read meaning from the form of a function — what its constants, factors, and terms represent — and connect an equation to a graph or a real-world situation.

Tested on SAT ≈ 6–8 questions per test · Advanced Math

What College Board tests

Quadratic and exponential functions, plus polynomial and rational functions, used to model relationships. The most common task is interpreting what a number or a function value means in context. You'll also find a function from given conditions, and identify the equation or graph that matches another.

Three forms of a quadratic

Every quadratic can be written three ways. Each form exposes one feature directly, so convert to the form that matches what the question asks for.

Standard
— gives the y-intercept directly.
Factored
— gives the x-intercepts r and s.
Vertex
— gives the vertex (h, k), the minimum or maximum.

The sign of sets the direction: opens up (minimum), opens down (maximum).

Exponential functions

An exponential function has the form . The starting value a is the output at , and the base b is the factor the output is multiplied by at each step. When the function grows; when it decays. If the rate applies over a period of several units rather than one, the exponent becomes , where is the length of that period.

Four worked examples in SAT format. Read the approach, try it yourself, then tap Show the full solution.

1 · Interpret a function value

The function gives the value, in dollars, of a savings account years after the account was opened, where . Which of the following is the best interpretation of in this context?

  1. The value of the account is estimated to be approximately $623 when the account is opened.
  2. The value of the account is estimated to be approximately $623 eight years after the account was opened.
  3. The value of the account is estimated to increase by approximately $623 each year for 8 years.
  4. The value of the account is estimated to be approximately $8 after 623 years.

Approach Translate the statement back into the situation. The input is years since opening and the output is dollars, so read as "the value at year 8."

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Answer: B

In , the input is the number of years since the account was opened and the output is the account's value in dollars. So means that when — eight years after opening — the value is about $623.

Why the other choices are wrong

A  Describes (account opened), not . At opening the value is .

C  Treats the change as a fixed yearly dollar increase. Exponential growth is multiplicative, not a constant amount.

D  Swaps the input and output — reads 623 as the year and 8 as the value.

2 · Find a function from given conditions

A quadratic function has a vertex at , and its graph passes through the point . What is the value of ?

Approach You're handed the vertex, so start in vertex form. Use the extra point to solve for , then evaluate at .

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Answer: C

Because the vertex is given, start in vertex form: . Substitute the point to find a:

So , and .

Why the other choices are wrong

A  Reports , the vertex's y-value, not .

B  Stops at without evaluating the function.

D  Sign error: computes instead of subtracting 4.

3 · Interpret a rate over an interval

The value, in dollars, of a piece of equipment is modeled by , where is the number of months since it was purchased. Which of the following is the best interpretation of this model?

  1. The value of the equipment decreases by 85% every 3 months.
  2. The value of the equipment decreases by 15% every 3 months.
  3. The value of the equipment decreases by 15% every month.
  4. The value of the equipment decreases by 15% every 3 years.

Approach Two pieces to read: the base sets the percent change (a base below 1 is a loss), and the exponent sets how often it happens — once every 3 of whatever unit measures.

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Answer: B

The base means each period the value is multiplied by 0.85 — it keeps 85% and so loses 15%. The exponent with in months means one period is 3 months. So the value drops 15% every 3 months.

Why the other choices are wrong

A  Confuses the base with the loss. A base of is a 15% loss, not an 85% loss.

C  Ignores the . The 15% drop happens every 3 months, not every month.

D  Misreads the unit of . It's months, so the period is 3 months, not 3 years.

4 · Read a transformed graph

Graph of y = f(x) passing through (2, 6) A curve rising left to right that passes through the point (2, 6). y x 2 6 2 6 (2, 6)

The graph of is shown. Which of the following is true about the graph of ?

  1. It passes through the point .
  2. It passes through the point .
  3. It passes through the point .
  4. It passes through the point .

Approach Subtracting outside the function, , shifts every point straight down by 3. The -values don't change.

Show the full solution

Answer: B

Subtracting 3 from the whole function, , shifts every point on the graph down 3 units. The -values don't change. Since the original graph passes through , the new graph passes through , or .

Why the other choices are wrong

A  Shifts up 3 instead of down: would give .

C  Shifts right 3. That's , a change inside the function, which moves the graph horizontally.

D  Shifts left 3. That's , again a horizontal change, not a vertical one.

Quick reference

Interpret a value
— translate input & output back into the situation.
Min / max
Vertex form — read .
X-intercepts
Factored form — read the zeros.
Growth / decay
Exponential — base is the multiplier; sets the period.
Shifts
moves up/down; moves left/right.
Shu's Tutoring SAT · Advanced Math · Nonlinear functions