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Math: Advanced Math, 3 original questions

SAT Nonlinear Functions Practice: Questions and Answers

Interpret vertices, exponential growth and nonlinear function parameters through three original SAT-style questions with full reasoning and choice explanations.

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The short answer

A nonlinear function does not change by the same amount for every equal change in its input. Identify the model and the meaning of its parameters. Quadratic vertex form reveals an extremum and its location; exponential form reveals an initial amount and a repeated growth factor. Distinguish an input from an output and a multiplier from a percentage. Use the stated context and domain when interpreting a function, rather than treating every coefficient as a rate of change.

Method at a glance

  1. 1Name the model and variables
  2. 2Read the helpful form
  3. 3Use the sign or factor
  4. 4Check with a simple input

Try the 3 questionsRead the full method

Step-by-step method

Use these steps in order, then apply them to the questions below.

  1. Step 1: Name the model and variables

    Recognize a quadratic, exponential or other nonlinear form. State what the input and output measure before interpreting parameters.

  2. Step 2: Read the helpful form

    In a(x − h)² + k, the vertex is (h, k). In A·b^t, A is the value at t = 0 and b is the factor per input interval.

  3. Step 3: Use the sign or factor

    A positive quadratic coefficient gives a minimum; a negative one gives a maximum. An exponential factor above 1 indicates growth, not a constant added amount.

  4. Step 4: Check with a simple input

    Use the vertex input or t = 0 to check your interpretation. Keep the input interval and units attached to the answer.

Try 3 SAT-style questions

These are original SAT-style questions written by PeakSAT, not official College Board questions. Choose an answer and check it to see why each choice is right or wrong, or open the worked answer.

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Question 1 of 3

For all real x, f(x) = (x − 4)² + 6. What is the minimum value of f(x)?

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Correct answer

Choice B: 6

Explanation

A square is at least zero. The term (x − 4)² reaches zero at x = 4, leaving f(4) = 6. All other inputs give a value at least 6, so 6 is the minimum output.

Why each choice is right or wrong

  1. Choice A

    This is the input where the minimum occurs, not the minimum output.

  2. Choice B

    Correct answer

    Correct: the square can be zero, leaving an output of 6.

  3. Choice C

    Adding the vertex coordinates does not give the minimum.

  4. Choice D

    This ignores the constant and substitutes the wrong value into the square.

Question 2 of 3

A model for a population is P(t) = 240(1.12)^t, where t is the number of years after the model starts. What does 1.12 represent?

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Correct answer

Choice C: An increase of 12% each year

Explanation

The population is multiplied by 1.12 for each one-year increase in t. Since 1.12 = 1 + 0.12, this model represents 12% annual growth. The initial population is 240, and the annual added number changes as the population changes.

Why each choice is right or wrong

  1. Choice A

    240 is the initial value, not the annual amount added.

  2. Choice B

    An exponential factor multiplies the current amount; it is not an added number of individuals.

  3. Choice C

    Correct answer

    Correct: the factor includes the original 100% plus an additional 12%.

  4. Choice D

    112% is the new amount as a percentage of the previous amount; the increase alone is 12%.

Question 3 of 3

For all real x, g(x) = −2(x + 3)² + 8. Which choice gives the vertex of the graph?

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Correct answer

Choice D: (−3, 8)

Explanation

The squared term becomes zero when x + 3 = 0, so x = −3. At that input, g(−3) = 8. The vertex is (−3, 8). Because −2 is negative, this vertex is also the maximum point.

Why each choice is right or wrong

  1. Choice A

    The expression x + 3 equals zero at −3, not 3.

  2. Choice B

    The constant output at the vertex is positive 8, not −8.

  3. Choice C

    Both coordinate signs are incorrect.

  4. Choice D

    Correct answer

    Correct: substituting x = −3 leaves the output 8.

Common mistakes

  • Confusing the vertex input with its output. A minimum value is a y-value, not the x-value where it occurs.

  • Reading 1.12 as 112% growth. It represents retaining 100% and adding 12% each interval.

  • Ignoring a negative leading coefficient. A downward-opening quadratic has a maximum at its vertex.

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Common questions

How do I distinguish linear from exponential change?

For equal input steps, linear models add a constant amount. Exponential models multiply by a constant positive factor.

Is the vertex always a minimum?

No. It is a minimum when the quadratic opens upward and a maximum when it opens downward.

Does an exponential model describe the situation forever?

Not necessarily. A mathematical model can be useful over a stated interval without remaining realistic outside that context or range.

Sources

  1. College Board: Advanced Mathsatsuite.collegeboard.org
  2. College Board: Student Question Bank Math skillssatsuite.collegeboard.org

Sources are linked for reference; they do not endorse PeakSAT. How we prepare our guides