Algebra
Quadratic equations and functions
Quadratic functions and graphs
20 practice questions
2 video lessons
Theory + worked examples
Theory
A quadratic function \(y=ax^2+bx+c\) graphs as a parabola:
- Vertex: the highest or lowest point.
- Axis of symmetry: \(x=-\dfrac{b}{2a}\).
- Opens up if \(a>0\), down if \(a<0\).
- \(y\)-intercept: \(c\).
The axis of symmetry passes through the vertex.
Vertex, axis of symmetry, and direction.
Features of a parabola.
Axis of symmetry:
\[x=-\dfrac{b}{2a}\]
Find the axis first, then the vertex.
How to graph a quadratic
- Find the axis of symmetry \(x=-\dfrac{b}{2a}\).
- Find the vertex on that axis.
- Plot the \(y\)-intercept \(c\).
- Use symmetry to plot more points.
Example 1 — Direction
Does \(y=-x^2+3\) open up or down?
Solution
\(a=-1<0\), so it opens down.
| \(a<0\) | \(\Rightarrow\) | \(\text{opens down}\) |
Example 2 — Axis of symmetry
Find the axis of symmetry of \(y=x^2-4x+1\).
Solution
Use \(x=-\dfrac{b}{2a}\).
| \(x\) | \(=\) | \(-\dfrac{-4}{2}=2\) |
Example 3 — Vertex
Find the vertex of \(y=x^2-4x+1\).
Solution
Axis \(x=2\); then \(y=4-8+1=-3\).
| \(\text{vertex}\) | \(=\) | \((2,-3)\) |
Example 4 — y-intercept
What is the \(y\)-intercept of \(y=x^2-4x+1\)?
Solution
It is the constant \(c\).
| \(y\) | \(=\) | \(1\) |
Common pitfalls
The axis is \(x=-\dfrac{b}{2a}\) — mind the sign.
\(a>0\) opens up, \(a<0\) opens down.
The vertex is on the axis of symmetry.
Frequently asked questions
What is the vertex?
The highest or lowest point of the parabola.
What is the axis of symmetry?
\(x=-\dfrac{b}{2a}\), through the vertex.
How do you know the direction?
Up if \(a>0\), down if \(a<0\).
What is the y-intercept?
The constant \(c\).
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