The formula this page uses
f(x) = a · g( b(x − h) ) + k
Precalculus Calculator
Solve precalculus problems with clear steps, notation, and a final check. Choose a focused operation below to calculate, inspect the method, and connect the result to a visual model.
The visual updates with your calculation.
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Calculus
Precalculus Calculator explained
The short version
- Precalculus is mostly about functions: what goes in, what comes out, and how the graph moves.
- Every transformation lives in one template, f(x) = a·g(b(x − h)) + k, and each letter does exactly one job.
- An inverse function undoes a function. It is not the reciprocal, even though both are written with a −1.
What each part means
SymbolWhat it means
a — Vertical stretchMultiplies every output. |a| > 1 stretches taller, 0 < |a| < 1 flattens, a negative flips it upside down.
b — Horizontal stretchDivides every input. |b| > 1 squeezes the graph toward the y-axis.
h — Horizontal shiftMoves the graph right by h. The sign inside the bracket is the opposite of the direction.
k — Vertical shiftMoves the graph up by k. This one behaves the way it looks.
Show your work: a full example
- The functionf(x) = −2(x − 3)² + 5
- Name the parentg(x) = x², a plain upward parabola with vertex (0, 0)
- Read h from inside the bracketx − 3 means h = 3, so the graph slides 3 to the right
- Read a in fronta = −2, so it stretches to twice as tall and flips upside down
- Read k on the endk = 5, so it lifts 5, putting the vertex at (3, 5)
- Evaluate at x = 1−2(1 − 3)² + 5 = −2(−2)² + 5 = −2(4) + 5 = −8 + 5 = −3
- State the rangethe parabola opens downward from (3, 5), so y ≤ 5
A second, different case
- A different case: build an inversef(x) = 2x − 7
- Write it as an equation in yy = 2x − 7
- Swap x and yx = 2y − 7
- Add 7 to both sidesx + 7 = 2y
- Divide both sides by 2y = (x + 7) / 2, so f⁻¹(x) = (x + 7) / 2
- Test it: run 5 through the inversef⁻¹(5) = (5 + 7) ÷ 2 = 12 ÷ 2 = 6
- Run that answer back through ff(6) = 2(6) − 7 = 12 − 7 = 5, so the two functions really do undo each other
Copy-ready example
f(x) = −2(x − 3)² + 5
The function
Questions about the Precalculus Calculator
Why is precalculus almost entirely about functions?
Because calculus asks how a function changes, and that question is meaningless until you can name a function's domain, read its graph and recognise its shape. Precalculus is where the vocabulary gets built.
How do I tell whether a function has an inverse?
Use the horizontal line test on its graph. y = 2x − 7 passes every horizontal line exactly once, so it has an inverse. y = x² fails, which is why √x is defined only for x ≥ 0 in the table above.
What is the difference between f(g(x)) and f(x) · g(x)?
Composition feeds one output into the other function, so f(g(2)) means work out g(2) first and hand the result to f. Multiplication evaluates both at the same x and multiplies the two heights. They almost never give the same number.
Do I need to be comfortable with radians before calculus?
Yes. Derivative rules like d/dx sin x = cos x are only true when x is in radians; in degrees an extra factor of π/180 appears. Getting used to π/6 and 5π/6 now saves rewriting every trig derivative later.
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