Quadratic Function

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Quadratic Function: meaning, method and solved example (Algebra 1) – Maths PDF

In short: Quadratic function is the name for any rule f(x) = ax² + bx + c with a ≠ 0; plotted, it always sweeps out a smooth bowl or arch.

Algebra 1Functions and GraphsF-IF.C.7a

What is Quadratic Function?

Give a rule an x-squared as its biggest power and you obtain a quadratic function, \(f(x) = ax^2 + bx + c\) with \(a \neq 0\). Instead of marching along a line, its graph swings into a bowl or an arch called a parabola.

The sign of \(a\) picks the orientation: positive gives a bowl with a lowest point, negative gives an arch with a highest point. The constant \(c\) is where the curve meets the y-axis, and an invisible vertical mirror runs through the turning point.

How to spot it

  • A quadratic function has x² as its biggest power; lower powers and a constant may tag along.
  • A graph bends once, like a bowl or an arch, with a single turning point.
  • A tossed ball, a suspension cable or a profit model peaks or bottoms out.
  • In a table with equal x-steps, the second differences stay constant while the first differences do not.

The method

  1. Read a, b and c, and let the sign of a tell you bowl or arch.
  2. The y-intercept is c, and the mirror line is \(x = -\frac{b}{2a}\).
  3. Feed that x back into the rule to learn the height of the turning point.
  4. Solve \(f(x) = 0\) for the x-intercepts, then plot the points and connect them smoothly.
vertex (2, -1)(1, 0)(3, 0)(0, 3)f(x) = x² − 4x + 3a = 1: opens upaxis of symmetry x = 2
f(x) = x² − 4x + 3: vertex (2, −1), zeros at 1 and 3, y-intercept 3.

Quadratic Function: solved example

Problem

Describe \(f(x) = x^2 – 6x + 8\): its orientation, y-intercept, turning point and x-intercepts.

Solution

Because \(a = 1\) the curve is a bowl, and \(c = 8\) puts the y-intercept at \((0, 8)\). The mirror line is \(x = -\frac{-6}{2(1)} = 3\), and \(f(3) = 9 – 18 + 8 = -1\), so the turning point is \((3, -1)\).

Factoring \((x – 2)(x – 4) = 0\) gives the x-intercepts \((2, 0)\) and \((4, 0)\).

Try it

Question 1

Does \(f(x) = -3x^2 + x\) make a bowl or an arch?

Show the answer

An arch, because \(a = -3\) is negative.

Question 2

Evaluate \(f(x) = 2x^2 – x + 5\) when \(x = -2\).

Show the answer

\(2(4) + 2 + 5 = 15\).

Quadratic Function vs. Linear Function

Straight-line rules keep x to the first power, so the steepness never changes. An x-squared term makes the slope shift continuously, and a turning point appears where the graph reverses. One draws a ruler line, the other draws a swing.

Linear Function

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