SAT Nonlinear Functions Questions
Quadratic, exponential and polynomial functions. Vertex, zeros, growth and decay, and reading all of it off a graph.
In short
- Nonlinear Functions is part of Advanced Math, about 35% of the scored Math questions: 13 to 15 on every test.
- Math questions from all four areas are mixed in each module and arranged from easiest to hardest.
- Vertex form is the trap Rivie's practice tests set most often in Nonlinear Functions questions.
Common traps
Most common first, from Rivie's ten practice tests.
Vertex form
Quadratic in vertex form a(x − h)² + k; the wrong choice mis-reads h as positive when it's negative (or vice versa) inside the parentheses.
The usual mistake: You see (x − 3) and read the vertex as h = −3 instead of h = 3.
Zero multiplicity behavior
A zero of even multiplicity touches the x-axis without crossing; odd multiplicity crosses. The wrong choice ignores this and treats all zeros the same.
The usual mistake: You read off the zeros from a graph and ignore whether the curve bounces or pierces, losing the distinction between (x − 2)² and (x − 2).
Shifted variable
Function transformation problems where the wrong choice moves the graph in the opposite direction (or applies horizontal shift inside the parentheses with the wrong sign).
The usual mistake: You see f(x − 3) and shift left because of the minus sign, instead of right.
Vertex: x vs. y coordinate
The vertex of a parabola y = a(x − h)² + k sits at (h, k). The wrong choice swaps which coordinate is asked for: giving y when the question wants x, or vice versa.
The usual mistake: You compute the vertex correctly but report the wrong axis when the question says 'what is the x-coordinate'. Slow down and re-read after you find the point.
Exponential growth rate
In y = a · (1 + r)^t, the growth rate is r as a decimal: a 5% rate is 0.05 and the multiplier is 1.05. The wrong choice picks the multiplier as the rate, or vice versa.
The usual mistake: You see 1.05 and report '105%' as the growth rate, missing that the model encodes growth as the increment ABOVE 1 (so 5%).
Linear vs. exponential
Tables/graphs that show equal differences between consecutive y-values are linear; tables that show equal RATIOS are exponential. The wrong choice picks the model that doesn't fit.
The usual mistake: You eyeball the trend (it goes up, fast) and pick exponential because it sounds dramatic, without checking whether the differences or the ratios are constant.
Try a question
From Rivie's practice tests. Pick an answer to see why it is right or wrong.
The function h is defined by h(x) = 2(x + 3)² − 5. What is the minimum value of h?
Video lessons
Free on YouTube, picked for this topic.