Advanced Topics
In a nutshell: Functions describe relationships, and graphs help you see them visually.
## Mapping Inputs to Outputs
A function is a rule that assigns each input exactly one output. Functions appear on the ACT as equations, tables, and graphs.
### Reading Graphs
- The \( x \)-axis is horizontal; the \( y \)-axis is vertical.
- The graph of \( y = mx + b \) is a straight line.
### Domain and Range
- Domain: All possible \( x \) values.
- Range: All possible \( y \) values.
### Real-World Application
Use functions to track things like speed, costs, or population growth over time.
### Key Skill
Interpreting function notation, like \( f(x) \), and evaluating for specific values.
### Example Formula
- Slope of a line: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
m = \frac{y_2 - y_1}{x_2 - x_1}
Examples
- If \( f(x) = 2x + 1 \), then \( f(3) = 7 \)
- The graph of \( y = x^2 \) is a parabola
Key terms
- Function
- A relationship that assigns exactly one output to each input.
- Domain
- All possible input values for a function.