SAT Math · Advanced Math · Skill 3 of 3

Nonlinear functions

Read quadratic and exponential functions from equations, tables, graphs and real situations. Know which form shows what, and turn “grows 6% a year” into a formula in seconds.

9 chapters plus a challengeAbout 40 minutesFree, no sign-up
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Hi, I’m Io. Parabolas and growth curves, read from every angle.
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Shape a parabola

Explore · 5 min

In vertex form, y = a(x − h)² + k, the turning point is (h, k). If a is positive the parabola opens up and k is the minimum value. If a is negative it opens down and k is the maximum. A bigger |a| makes it narrower.

Three forms, three stories

Game · 4 min

The same quadratic can be written three ways. Each form shows one feature as a number you can read straight off:

  • Standard y = ax² + bx + c: the y-intercept is c
  • Vertex y = a(x − h)² + k: the vertex is (h, k)
  • Factored y = a(x − p)(x − q): the x-intercepts are p and q
Sort by form

Reading key features

Practise · 5 min

A parabola is symmetric, so its vertex sits on the axis of symmetry, halfway between the x-intercepts. Find that x-value, then substitute it to get the minimum or maximum.

Worked example: f(x) = (x − 1)(x − 7)
  1. x-intercepts: 1 and 7
  2. Axis of symmetry: halfway between, so x = 4
  3. Minimum value: f(4) = (3)(−3) = −9, so the vertex is (4, −9)
  4. y-intercept: f(0) = (−1)(−7) = 7

Exponential growth and decay

Learn · 5 min

An exponential function has the form f(t) = a · bt. The starting value is a. The growth factor b is what you multiply by each time period.

  • Grows 6% a year: b = 1 + 0.06 = 1.06
  • Loses 15% a year: b = 1 − 0.15 = 0.85
  • Doubles every period: b = 2. Halves: b = 0.5

Linear or exponential?

Game · 4 min

Check what happens from one step to the next. Same difference each time: linear. Same ratio each time: exponential. In words, a fixed amount is linear and a fixed percent is exponential.

Function notation and shifts

Practise · 4 min

f(4) means “put 4 in for x”. Changes to f move its graph:

  • f(x) + k moves the graph up k. f(x) − k moves it down.
  • f(x − h) moves the graph right h. f(x + h) moves it left.

Real-life models

Scenarios · 6 min

SAT questions often give a model and ask what one of its numbers means. The starting value, the growth factor and the vertex each tell part of the story.

Read it in Desmos

Tools · 3 min

Desmos marks the vertex, intercepts and any point you tap. It is the fastest way to find a maximum height or a value after a number of years.

Spot the trap

Detective · 3 min

Each solution has one faulty step. The first and third are the two most common slips with nonlinear functions.

Final challenge

8 questions · 10 min

Key ideas to carry in
Vertex forma(x − h)² + k has vertex (h, k).
SymmetryThe vertex is halfway between the x-intercepts.
Growth factorUp r%: 1 + r. Down r%: 1 − r.
Linear or exponentialSame difference or same ratio.
Shiftsf(x − h) moves right. f(x) + k moves up.
ModelsRead what each number means in the story.
Challenge