Algebra
Relations and functions
Key features of graphs
20 practice questions
2 video lessons
Theory + worked examples
Theory
The key features of a graph describe its shape:
- \(x\)-intercepts (zeros): where it crosses the \(x\)-axis.
- \(y\)-intercept: where it crosses the \(y\)-axis.
- Maximum / minimum: highest or lowest points.
- Increasing / decreasing intervals.
Read features directly from the graph — intercepts, turning points, and direction.
Intercepts, maximum, and direction.
The key features.
Intercepts:
\[\text{x-intercept: } f(x)=0,\qquad \text{y-intercept: } f(0)\]
Zeros and \(x\)-intercepts are the same thing.
How to read key features
- Find where the graph meets each axis.
- Locate the highest/lowest point.
- Note where it rises and falls.
- Describe each feature.
Example 1 — Intercepts
A graph crosses the \(x\)-axis at \(-2\) and \(3\), the \(y\)-axis at \(2\). Name the intercepts.
Solution
Read where it meets each axis.
| \(\text{x-intercepts}\) | \(=\) | \(-2,\ 3\) |
| \(\text{y-intercept}\) | \(=\) | \(2\) |
Example 2 — Maximum
A parabola opens down with vertex \((1,5)\). What is its maximum?
Solution
The vertex is the highest point.
| \(\text{maximum}\) | \(=\) | \(5\ \text{at}\ x=1\) |
Example 3 — Increasing interval
An upward parabola has vertex at \(x=2\). Where is it increasing?
Solution
It rises to the right of the vertex.
| \(\text{increasing}\) | \(\text{on}\) | \((2,\infty)\) |
Example 4 — Zeros
What are the zeros of a function?
Solution
The zeros are the \(x\)-intercepts — where \(f(x)=0\).
Common pitfalls
Zeros are \(x\)-intercepts, where \(f(x)=0\).
There is one \(y\)-intercept for a function.
Read direction left to right.
Frequently asked questions
What are the key features of a graph?
Intercepts, maximum/minimum, and intervals of increase/decrease.
What is a zero of a function?
An \(x\)-intercept — where \(f(x)=0\).
How many y-intercepts can a function have?
Exactly one.
What is a maximum?
The highest output value on the graph.
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