SAT Math · Lesson 2 of 4
Advanced math: nonlinear relationships
Work with quadratics, equivalent forms, exponents, and functions when the relationship is no longer a straight line.
Let the form do the work
A quadratic can be written in several forms. Standard form ax² + bx + c is useful for a y-intercept. Factored form shows the roots. Vertex form a(x − h)² + k shows the maximum or minimum at x = h. Choose the form that displays the quantity the question asks for.
If f(x) = 2(x − 3)² + 5, the squared term is never negative, so the smallest value of f is 5, reached at x = 3. Expanding would hide that minimum.
Functions, exponents, and restrictions
Evaluating f(4) asks for an output. Solving f(x) = 4 asks for the inputs that produce 4. Those are different tasks. A function machine can have two inputs with the same output, but each input has only one output.
With exponents, (x²)³ = x⁶, while x² × x³ = x⁵. A square root or rational expression may exclude values. If a denominator is x − 2, x = 2 is not allowed even if a later simplification no longer shows that factor.
Worked example
The function g(x) = −(x + 1)² + 9. What is the maximum value of g?
- (x + 1)² is at least 0 for every real x, and it is 0 when x = −1.
- Multiplying by −1 turns that minimum into a maximum, so −(x + 1)² is at most 0.
- Adding 9 gives a maximum of 9.
Why this works. The sign of a tells you whether the vertex is a maximum or a minimum. The k in vertex form is that extreme value.
Check your understanding
- Match the requested feature — root, vertex, or intercept — to a form.
- Distinguish f(a) from solving f(x) = a.
- Keep excluded values when a radical or denominator is present.