Interpret quadratic, exponential, and other nonlinear functions from equations, graphs, and tables.
You read information out of nonlinear functions rather than just solving them. Quadratic questions ask for a vertex, a minimum or maximum value, a line of symmetry, or which form of the equation shows a feature most directly. Exponential questions ask for a growth or decay rate, a starting value, or a future output given two known outputs. You also see function notation with transformations, piecewise definitions, and questions that give a graph and ask which equation could produce it.
Nonlinear functions are one of the heaviest Advanced Math topics, usually around 3 to 5 of the 44 math questions.
Written for this skill, with the answer explained. 322 more are in the app.
The graph of the quadratic function f is shown in the xy-plane. What is the y-coordinate of the vertex of the graph?
A population doubles every 3 years. If it is 1000 now, what will it be in 12 years?
Why: Find the number of doubling periods: 12/3 = 4. Each period multiplies the population by 2, so multiply by 2^4 = 16, giving 1000 * 16 = 16000. Answering 4000 multiplies by the number of doublings, 4, instead of by 2^4. Answering 8000 uses only 3 doublings, 1000 * 2^3. Answering 32000 uses 5 doublings, 1000 * 2^5.
A ball is thrown upward and its height in feet after t seconds is h(t) = -16t² + 64t. What is the maximum height of the ball, in feet?
Why: The maximum is at the vertex, t = -64/(2 · -16) = 2 seconds. Evaluate there: h(2) = -16(4) + 64(2) = -64 + 128 = 64 feet. The value 2 reports the time at which the maximum occurs, not the height. The value 48 is h(1), the height after one second, which is before the ball reaches the top. The value 32 is a height the ball passes through on the way up, since -16t² + 64t = 32 at t = 2 - √2, and the ball keeps rising after that.
A machine loses 20% of its value every year. What percent of its original value remains after 3 years?
Why: Each year leaves 80%, so after 3 years 0.8³ = 51.2% remains. The value 40 removes 20% of the original value three times, which decays too fast. Answering 60 means you multiplied 20 by the 3 years. Answering 64 means you used 2 years instead of 3.
Rewrite the function in vertex form, f of x equals a times the quantity x minus h squared plus k. If a is positive, the minimum value is k and it happens at x equals h. If a is negative, k is the maximum instead.
Look at how the output changes as the input steps up by one. If it changes by a constant amount added each time, the model is linear. If it changes by a constant multiplier each time, the model is exponential.
It is the growth factor per period. A base greater than 1 means growth, and subtracting 1 gives the percent increase. A base between 0 and 1 means decay, and subtracting it from 1 gives the percent decrease.
Questions on nonlinear functions alone, with explanations on every one, and the app tracks which ones you keep missing.
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