Functions
Functions describe how one quantity determines another, and the Regents Algebra I exam tests function notation, domain and range, average rate of change, graph features, and families of functions. You should move among equations, tables, graphs, and verbal descriptions. Linear, quadratic, and exponential models are especially important, along with transformations and sequences as functions with integer inputs.
Core rules
- f(a) means substitute a for every x. For f(x+h), replace x with the entire expression x+h; do not add h to f(x).
- Average rate of change on [a,b] is (f(b)−f(a))/(b−a). It is a slope between two points, not necessarily the slope at one point.
- Linear functions have a constant rate of change; exponential functions have a constant percent change or multiplier. A growth factor of 1.07 means a 7% increase.
- Transformations: f(x)+k shifts vertically by k; f(x+h) shifts horizontally opposite the sign of h; a·f(x) stretches or compresses vertically and may reflect across the x-axis.
- Domain restrictions come from context and from operations, such as division by zero.
Worked example
For f(x)=x^2−4x+1, find the average rate of change from x=1 to x=4. Compute f(4)=16−16+1=1 and f(1)=1−4+1=−2. Then (1−(−2))/(4−1)=3/3=1. The function rises by 1 unit per unit on average over that interval.
Traps
Do not confuse f(x+h) with f(x)+h. Do not call an average rate of change the slope at a single point. In exponential models, distinguish the percent rate from the growth factor. Check whether a graph's domain is limited by the real-world situation, such as time not being negative.