Nonlinear Functions

Choose the quadratic form that reveals the requested feature

The function $f(x)=2(x-3{)}^2-8$ has vertex $(3,-8)$. Because the coefficient 2 is positive, $-8$ is its minimum output. To find zeros, set it equal to zero: $(x-3{)}^2=4$, giving $x=1$ and $x=5$.

In factored form, $f(x)=2(x-1)(x-5)$ makes those zeros visible. The vertical intercept is found separately: $f(0)=10$. A root is an input; a minimum value is an output.

Use the axis of symmetry as a check

The zeros 1 and 5 are equally far from 3, so the axis is $x=3$. For a quadratic in standard form $a{x}^2+bx+c$, the axis is $x=-\frac{b}{2a}$. Substitute that input to find the vertex's output.

Completing the square can rewrite a standard-form expression when a question asks for the minimum or maximum. Keep the leading coefficient attached to the entire squared term.

Read exponential multipliers over the stated interval

In $P(t)=200(1.08{)}^t$, where $t$ counts years, 200 is the initial amount and 1.08 means an 8% increase each year. For $200(0.92{)}^t$, the amount decreases by 8% per year.

If the exponent is $t/3$, the multiplier applies every three years. The annual multiplier is $(1.08{)}^{1/3}$, not 1.08. Read the input unit and exponent together.

Separate transformations from individual values

Changing $f(x)$ to $f(x)+4$ raises every output by 4. Changing it to $f(x-4)$ shifts its graph right by 4. For the vertex expression $(x-h{)}^2$, the input reaches the vertex when $x=h$.

Test a transformed expression at a convenient input if the direction is unclear. That check is more reliable than trying to remember a sign rule in isolation.

Check your understanding

Sample nonlinear functions questions

Practice this skill

1. Identify whether the prompt asks for an input, output, rate, or parameter.
2. Rewrite only when the current form hides the requested feature.
3. Check graph readings with a substitution or symmetry calculation.

Print the nonlinear functions worksheet and worked answers.

Then use a mixed practice module to check whether you can choose this method among other question types.