Key takeaway: A quadratic function is any function whose rule can be written f(x) = ax² + bx + c, provided a is nonzero, and plotting it always produces a parabola.
Grade 9 – Algebra 1F-IF.C.7aFunctions and Graphs
Definition of Quadratic Function
A quadratic function has an \(x^{2}\) term as its highest power. Its output does not follow a straight-line pattern: as \(x\) increases, the output falls and then rises (or rises and then falls), turning around exactly once.
Its picture is a smooth bowl-shaped curve, a parabola. If \(a > 0\) it opens upward and has a lowest point; if \(a < 0\) it opens downward and has a highest point. That turning point is the vertex. Quadratics describe the path of a thrown ball, the area of a square as its side grows, and the profit of a business as the price changes.
Formula and notation
The standard form of the rule is
\[f(x) = ax^{2} + bx + c, \qquad a \neq 0\]
The number \(c\) is the \(y\)-intercept, because \(f(0) = c\). If \(a\) were 0, the \(x^{2}\) term would vanish and the rule would become linear.
Key facts
- Each graph has one vertex and a vertical mirror line.
- The sign of a decides whether the curve opens up or down.
- A quadratic function can have 0, 1 or 2 x-intercepts.
- Second differences of equally spaced outputs are constant, unlike linear patterns.
Quadratic Function example
For \(f(x) = x^{2} – 4x – 5\), tell which way the graph opens and find the \(y\)-intercept, the zeros and the vertex.
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Here \(a = 1 > 0\), so the curve opens upward. The \(y\)-intercept is \(f(0) = -5\).
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Factor: \(x^{2} – 4x – 5 = (x – 5)(x + 1)\), and the output is 0 when \(x = 5\) or \(x = -1\).
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The vertex lies halfway between the zeros at \(x = 2\), and \(f(2) = 4 – 8 – 5 = -9\).
It opens upward, with \(y\)-intercept \(-5\), zeros \(-1\) and 5 and vertex \((2, -9)\).
Common mistake
Watch out
Looking at \(g(x) = 2x(x + 3)\), Eli says it is not quadratic because he cannot see an \(x^{2}\) term.
Why it is wrong
The \(x^{2}\) term is hidden by the factored form. Distribute: \(g(x) = 2x^{2} + 6x\), which has \(a = 2\), \(b = 6\), \(c = 0\). Multiplying out reveals the true degree of any factored rule.
Related terms and lessons
- Parabola
- Vertex of a Parabola
- Axis of Symmetry
- Standard Form of a Quadratic
- Zeros of a Function
- Quadratic Formula
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