3.1 Functions and Graphs
Key Takeaways
- Function notation means substitute the input into the rule.
- Slope is vertical change divided by horizontal change.
- Domain restrictions usually come from denominators, even roots, and logarithms.
- Quadratic graphs are controlled by vertex, intercepts, and opening direction.
Read the Function First
Function notation is a common placement trap. The expression f(3) means substitute 3 into the rule for x. It does not mean multiply f by 3. If f(x) = 2x - 5, then f(3) = 1.
Graph Basics
| Feature | How to find it |
|---|---|
| Slope | (y2 - y1) / (x2 - x1) |
| y-intercept | Set x = 0 |
| x-intercept | Set y = 0 |
| Domain | List allowed inputs |
| Range | List possible outputs |
| Vertex | Turning point of parabola |
Restrictions to Memorize
Denominators cannot equal zero. Even roots cannot have negative radicands when working in real numbers. Logarithm inputs must be positive. These three checks solve many domain questions before any graphing is needed.
For line questions, identify whether you are being asked for slope, an intercept, an equation, or a parallel/perpendicular relationship. Parallel lines use the same slope. Perpendicular nonvertical lines use negative reciprocal slopes.
Graph Practice
Alternate between algebraic and visual versions of each skill. Find intercepts from an equation, then read intercepts from a graph. Calculate slope from two points, then estimate slope from a plotted line. This prepares you for ALEKS items that change representation without changing the underlying concept.
If f(x) = 3x + 2, what does f(4) mean?