Cubic Functions Of The Formf(X) =A(X−H)3+K
Understand transformed cubics of the form a times x minus h cubed plus k for Queensland Year 11 Mathematical Methods (QCAA). This rule shifts, stretches and reflects the basic cubic curve.
You will learn to read the stationary point of inflection, describe the effect of the parameters a, h and k, find the single x-intercept, and note the behaviour for large positive and negative x — core QCAA graphing.
Every question with a fully worked solution.
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Theory
In Year 11 Mathematical Methods (QCAA, Unit 1), the cubic \(y=a(x-h)^3+k\) is the graph of \(y=x^3\) after a dilation, a possible reflection, and translations. Its stationary point of inflection is \((h,k)\). This page covers reading \((h,k)\), the effect of \(a\), the transformations from \(y=x^3\), the intercepts, and finding the rule from features — all without calculus.
The graph of \(y=x^3\) is the basic cubic: it increases everywhere and has a stationary point of inflection at the origin, where it briefly flattens before continuing. It has no turning points.
The transformed cubic \(y=a(x-h)^3+k\) keeps this shape. The inflection point moves to \((h,k)\): \(h\) shifts it horizontally and \(k\) vertically. The number \(a\) is a dilation that changes steepness, and if \(a\lt 0\) it reflects the curve in the \(x\)-axis so it decreases everywhere.
Because the cube is a one-to-one function, \(y=a(x-h)^3+k\) has exactly one \(x\)-intercept, found by taking a cube root (which always gives one real value).
The transformed cubic and its inflection point:
The \(y\)-intercept (set \(x=0\)):
The single \(x\)-intercept (set \(y=0\), then cube-root):
How to work with \(y=a(x-h)^3+k\)
- Inflection: read \((h,k)\) directly from the rule, watching the sign of \(h\).
- Shape: use \(a\) for steepness and direction — \(a\gt 0\) increasing, \(a\lt 0\) decreasing (reflected) — and state the end behaviour.
- Intercepts / rule: put \(x=0\) for the \(y\)-intercept and \(y=0\) then cube-root for the \(x\)-intercept; or, given the inflection and one point, substitute to solve for \(a\).
Comparing with \(y=a(x-h)^3+k\) gives \(a=1,\ h=2,\ k=1\), so the inflection is \((2,1)\).
\(y\)-intercept — substitute \(x=0\):
| \(y\) | \(=\) | \((0-2)^3+1\) |
| \(=\) | \(-8+1\) | |
| \(=\) | \(-7\) |
\(x\)-intercept — set \(y=0\) and cube-root:
| \((x-2)^3+1\) | \(=\) | \(0\) |
| \((x-2)^3\) | \(=\) | \(-1\) |
| \(x-2\) | \(=\) | \(-1\) |
| \(x\) | \(=\) | \(1\) |
Inflection \((2,1)\); \(y\)-intercept \((0,-7)\); \(x\)-intercept \((1,0)\).
Reading the constants: \(a=-2,\ h=3,\ k=4\).
Interpret each constant:
| \(a=-2\) | \(=\) | \(\text{dilate by factor }2,\ \text{reflect in the }x\text{-axis}\) |
| \(h=3\) | \(=\) | \(\text{translate right }3\) |
| \(k=4\) | \(=\) | \(\text{translate up }4\) |
The inflection point is \((3,4)\). Because \(a\lt 0\) the curve decreases, so as \(x\to\infty,\ y\to-\infty\) and as \(x\to-\infty,\ y\to\infty\).
Dilation factor \(2\), reflection in the \(x\)-axis, then right \(3\) and up \(4\); inflection \((3,4)\), decreasing.
Here \(h=-1,\ k=-4\), so the inflection is \((-1,-4)\).
\(y\)-intercept — substitute \(x=0\):
| \(y\) | \(=\) | \((0+1)^3-4\) |
| \(=\) | \(1-4\) | |
| \(=\) | \(-3\) |
\(x\)-intercept — set \(y=0\) and cube-root:
| \((x+1)^3-4\) | \(=\) | \(0\) |
| \((x+1)^3\) | \(=\) | \(4\) |
| \(x+1\) | \(=\) | \(\sqrt[3]{4}\) |
| \(x\) | \(=\) | \(\sqrt[3]{4}-1\) |
\(y\)-intercept \((0,-3)\); \(x\)-intercept \(\left(\sqrt[3]{4}-1,\,0\right)\).
The inflection gives \(h=2,\ k=3\), so \(y=a(x-2)^3+3\).
Substitute the point \((0,-1)\) to find \(a\):
| \(-1\) | \(=\) | \(a(0-2)^3+3\) |
| \(-1\) | \(=\) | \(-8a+3\) |
| \(-8a\) | \(=\) | \(-4\) |
| \(a\) | \(=\) | \(\dfrac{1}{2}\) |
\(x\)-intercept of \(y=\dfrac{1}{2}(x-2)^3+3\) — set \(y=0\):
| \(\dfrac{1}{2}(x-2)^3+3\) | \(=\) | \(0\) |
| \((x-2)^3\) | \(=\) | \(-6\) |
| \(x-2\) | \(=\) | \(-\sqrt[3]{6}\) |
| \(x\) | \(=\) | \(2-\sqrt[3]{6}\) |
Rule \(y=\dfrac{1}{2}(x-2)^3+3\); \(x\)-intercept \(\left(2-\sqrt[3]{6},\,0\right)\).
Common pitfalls
Frequently asked questions
What is the point of inflection of \(y=a(x-h)^3+k\)?
It is the stationary point of inflection at \((h,k)\), read straight from the rule. Watch the sign: \((x+1)^3\) gives \(h=-1\).
How many x-intercepts does \(y=a(x-h)^3+k\) have?
Exactly one, because solving it needs a cube root, which always gives a single real value.
What does the value of \(a\) do to the graph?
It is a dilation that changes the steepness, and if \(a\) is negative it reflects the curve in the \(x\)-axis so the graph decreases instead of increases.
How do you find the x-intercept exactly?
Set \(y=0\), isolate \((x-h)^3=-\dfrac{k}{a}\), then cube-root: \(x=h+\sqrt[3]{-\dfrac{k}{a}}\), leaving the cube root exact.
How do you find the rule from the inflection and a point?
Read \(h\) and \(k\) from the inflection point, substitute the given point into \(y=a(x-h)^3+k\), and solve for \(a\).
Does \(y=a(x-h)^3+k\) have turning points?
No. It increases everywhere (or decreases if \(a\lt 0\)) with just a stationary point of inflection at \((h,k)\); there is no maximum or minimum.