Math101 · Independent Ontario learning library
Math101Quadratic Function
A quadratic function has constant nonzero second differences and graphs as a parabola whose forms reveal zeros, vertex, and intercepts.
A quadratic function can be written $f(x)=ax^2+bx+c$ with $a\ne0$. Its graph is a parabola.
Recognizing quadratic behaviour
The highest power is $2$. For equally spaced inputs, first differences change, but second differences are constant. With step size $1$, the constant second difference is $2a$.
Quadratic growth is not proportional: doubling $x$ does not generally double the output.
Three useful forms
| Form | Expression | Feature shown |
|---|---|---|
| standard | $a x^2+bx+c$ | $y$-intercept $c$ |
| vertex | $a(x-h)^2+k$ | vertex $(h,k)$ |
| factored | $a(x-r_1)(x-r_2)$ | zeros $r_1,r_2$ |
Equivalent forms describe the same function. Choose the form matching the question.
Vertex and axis
The axis of symmetry is
Substitute this $x$-value into $f$ to find the vertex. If $a>0$, the vertex is a minimum; if $a<0$, it is a maximum.
Intercepts and zeros
The $y$-intercept is $(0,c)$. The $x$-intercepts solve $f(x)=0$. Factoring, completing the square, or the quadratic formula may be used.
For $x^2-6x+5=(x-1)(x-5)$, zeros are $1$ and $5$. Their midpoint is $3$, matching the axis of symmetry.
Common mistakes
Calling any $x^2$ equation a quadratic function without checking $a\ne0$. A zero leading coefficient changes the degree.
Using $-b/2a$ as the vertex’s $y$-value. It is the $x$-coordinate.
Assuming every quadratic has two real zeros. It may have two, one, or none.
Mixing features from non-equivalent forms. Convert carefully and verify.
Quick self-check
- Which form makes the desired feature visible?
- What do sign and magnitude of $a$ imply?
- Are vertex, axis, and zeros symmetric?
- Does the domain match the context?
