A quadratic function has the form $f(x) = ax^2 + bx + c$ with $a \ne 0$: a polynomial of degree $2$, like the products you made when multiplying binomials. Its graph is a parabola, a U-shaped curve, and you can read it with the key features you already know.
Why it works
Start with $y = x^2$. Opposite inputs give the same output, $(-3)^2 = 3^2 = 9$, so the graph is symmetric about the $y$-axis, and it has a lowest point at $(0, 0)$. Multiplying by $a$ stretches every output: if $|a| > 1$ the outputs grow faster, so the graph looks narrower; if $|a| < 1$ it looks wider; and a negative $a$ flips it upside down. Every parabola keeps that symmetry about a vertical line, the axis of symmetry, through its turning point, the vertex.
- Opens up if $a > 0$ (the vertex is a minimum); opens down if $a < 0$ (the vertex is a maximum).
- The axis of symmetry is the vertical line through the vertex; points at the same height are mirror images across it.
- The $y$-intercept is $(0, c)$, since $f(0) = c$.
- The $x$-intercepts (zeros), if any, are where $f(x) = 0$.
Key idea
The sign of $a$ says up or down, $|a|$ says narrow or wide, $c$ is the $y$-intercept, and the axis of symmetry passes through the vertex, halfway between any two points at the same height.
Worked examples
Worked example: Read the features
Describe $y = -x^2 + 2x + 3$.
- $a = -1 < 0$: it opens down, so it has a maximum.
- $y$-intercept: $(0, 3)$.
- Two points at height $3$: $x = 0$ and, by testing, $x = 2$ (since $-4 + 4 + 3 = 3$). The axis is halfway: $x = 1$.
- Vertex: $y = -1 + 2 + 3 = 4$ at $x = 1$, so $(1, 4)$.
Answer: Opens down, vertex $(1, 4)$, axis $x = 1$, $y$-intercept $3$.
Worked example: Symmetry
A parabola passes through $(-1, 6)$ and $(5, 6)$. Find its axis of symmetry.
- Same height means mirror images. Halfway between $-1$ and $5$ is $\frac{-1 + 5}{2} = 2$.
Answer: $x = 2$
Common mistake
Thinking a larger $a$ makes the parabola wider. It does the opposite: in $y = 3x^2$ every output is three times as big as in $y = x^2$, so the graph climbs faster and looks narrower. Check one point: at $x = 1$, $y = 3x^2$ is already at height $3$.