Math101 · Independent Ontario learning library
Math101learn.math101.caQuadratic 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.
Opening and vertical stretch
The sign of $a$ determines opening. The magnitude $|a|$ affects vertical stretch: larger $|a|$ produces faster vertical change and a visually narrower graph; $0<|a|<1$ produces a wider graph.
The word “width” is visual shorthand—the parabola continues without a fixed width.
Domain and range
Every quadratic polynomial has domain all real numbers. Range begins or ends at the vertex value:
- if $a>0$, $y\ge k$;
- if $a<0$, $y\le k$,
where $(h,k)$ is the vertex.
Context may restrict domain, such as nonnegative time during a projectile’s flight.
Sketching efficiently
Identify opening, vertex, axis, intercepts, and one or two symmetric point pairs. Plot features before drawing a smooth curve. A parabola is not two straight line segments.
Modelling
Quadratics model constant acceleration, rectangular area, revenue under linear price-demand assumptions, and other situations with changing first differences. Interpret vertex and zeros in context, then reject values outside the feasible domain.
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?
Related topics
Explore the idea
Function transformation
Change one quantity at a time and connect what moves to Quadratic Function.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
What is the vertex of y = x² − 6x + 5?
- x = −(−6)/(2 · 1) = 3
- y = 3² − 6(3) + 5
- y = −4, so the vertex is (3, −4).
End of lesson
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