Rate of Change and Function Behavior

The rate of change (ROC) tells us how quickly a function's output changes compared to its input. We calculate it using the formula: ROC = $y₂ - y₁$/$x₂ - x₁$. This is essentially the slope of a line between two points.

Different functions have different ROC patterns. For linear functions, ROC is constant. For quadratic functions, ROC changes linearly. Understanding these patterns helps you identify function types from data.

When analyzing graphs, remember that positive ROC means the function is increasing, while negative ROC means it's decreasing. The concavity matters too—concave up shows an increasing rate of change, while concave down shows a decreasing rate.

Quick Tip: To find a polynomial's degree from a table of values, count how many times you need to take differences until you get a constant value. For example, if you need to take differences twice, it's a quadratic (degree 2).

Polynomials have special properties worth noting. The degree tells you the maximum number of zeros $x-intercepts$ possible. Even multiplicity zeros $like (x-2)²$ create tangent points with the x-axis, while odd multiplicity zeros $like (x-2)$ cross through the axis.

Polynomial and Rational Function Behavior

The end behavior of polynomials depends on the leading coefficient (LC) and the degree. For even-degree polynomials with positive LC, both ends point upward. Odd-degree polynomials with positive LC have one end up and one down.

Rational functions have special features called asymptotes. When the numerator's degree is less than the denominator's, you get horizontal asymptotes (HA). Equal degrees give you a horizontal asymptote at the ratio of leading coefficients. When the numerator's degree exceeds the denominator's by exactly 1, you get a slant asymptote.

Finding zeros of rational functions means setting the numerator to zero (assuming no holes). Holes occur when the same factor appears in both numerator and denominator. To find a hole, remove the common factor, then calculate the y-value at that point.

Remember: Vertical asymptotes (VA) occur where the denominator equals zero (without being canceled by the numerator). These are places where the function shoots toward infinity!

To find slant asymptotes, use polynomial long division and ignore the remainder term. This gives you the equation of the line the function approaches as x gets very large or very small.

Limits and Function Transformations

When working with rational functions, holes represent points where the function isn't defined, but the limit exists. We can approach these points from either direction:

• Left-hand limit: lim(x→a⁻) f(x)
• Right-hand limit: lim(x→a⁺) f(x)

For a hole, these limits are equal, though the function itself is undefined at that point.

Function transformations follow a specific pattern in y = af$b(x+c)$+d:

• a: Vertical dilation $stretches or compresses the y-values$
• b: Horizontal dilation by a factor of 1/b $affects x-values$
• c: Horizontal translation (shifts left or right)
• d: Vertical translation (shifts up or down)

Pro Tip: When transforming functions, work from inside to outside: first handle the horizontal shift (c), then horizontal dilation (b), followed by vertical dilation (a), and finally vertical shift (d).

These transformations let you create complex function shapes from simpler ones. Master them, and you'll be able to sketch almost any function quickly!