Key Terms
Real-world use
Average cost per item, drug concentration over time, mixing problems. Any time you divide one changing quantity by anoth
Process
1. Factor numerator and denominator.
Example
F(x) = sqrt((x + 2)(x - 3) / (x - 1)) Sign changes at x = -2, 1, 3. Domain where expression >= 0: [-2, 1) union [3, infi
Step-by-step
1. Find the y-intercept: evaluate f(0).
When reading a graph to write the function
1. Identify x-intercepts.
General form
F(x) = a * (x - x1)^p1 * (x - x2)^p2 * ... / (x - v1)^q1 * (x - v2)^q2 * ...
To verify
Compose both ways and confirm you get x each time.
Notation
F^(-1)(x) means the inverse of f, NOT 1/f(x).
FINDING k - PROCESS (all three types)
1. Write the general formula based on the variation description.
Example usage
As x -> 0-, f(x) -> -infinity Translation: as x approaches zero from the left, the output drops to negative infinity.
Vertical asymptote
A vertical line x = a where the graph tends toward positive or negative infinity as inputs approach a.
Arrow notation
Symbolic shorthand using arrows to describe how inputs or outputs behave as they approach a specific value or infinity.
Constant of variation
The nonzero value k that defines the proportional relationship in direct, inverse, or joint variation.
Direct variation
A relationship where y = k * x^n; as one quantity increases, the other increases proportionally.
Horizontal asymptote
A horizontal line y = b that the graph approaches as inputs increase or decrease without bound.