AP Precalc 1.1 Change in TandemAP Precalc 1.1 Change in Tandem

A P P R E C A L C U L U S · UNIT1

Change in Tandem

Topic 1.1 Study Guide

In one sentence: a function pairs every input with exactly one output, and we describe how the output changes as the input changes (increasing, decreasing, concave up, concave down). Function basics

Function: a relation that maps each input value to exactly one output value.

Input = domain = independent variable (x). Output = range = dependent variable (y). Notation h(t) names the output first and the input in parentheses: h = height, t = seconds. Four representations, VANG: Verbal, Analytic (formula), Numerical (table), Graphical. Example. w(x) = gallons of water in a pool, x = length of the pool. Independent: length of the pool. Dependent: gallons of water. The output is the thing that "depends on" the other. Increasing and decreasing

Verbal Analytic (on the interval)

Increasing As inputs increase, outputs always increase a < b ⇒ f(a) < f(b)

Decreasing As inputs increase, outputs always decrease a < b ⇒ f(a) > f(b)

Table example. x: 4, 6, 7, 10, 20 and f(x): 1, 1.01, 1.04, 1.06, 1.29. Every time x goes up, f goes up, so f is increasing on 4 ≤ x ≤ 20. Always justify with the "if a < b then …" sentence. Parts of a graph

Zero: an input where the output is 0 (graph meets the x-axis). Also called x-intercept or root. y-intercept: the output when the input is 0, that is f(0).

Concave up: graph bends like a cup; the rate of change is increasing.

Concave down: graph bends like a frown; the rate of change is decreasing.

Point of inflection: the point where concavity changes. Straight lines have no concavity. AP Precalculus Unit 1 · 1.1 1 of 2

Increasing/decreasing and concavity are independent

Shape Slopes are… Looks like

Increasing, concave up positive and getting bigger steeper and steeper climb

Increasing, concave down positive and getting smaller climb that levels off

Decreasing, concave up negative and getting closer to 0 drop that levels off

Decreasing, concave down negative and getting more negative steeper and steeper drop Traps. (1) A max or min is not an inflection point; inflection is where the bend flips. (2) "Concave" is about the bend, not whether the graph goes up or down. (3) Report intervals using x-values, never y- values. (4) Zeros are x-values; the y-intercept is a y-value.

Test-prep patterns

"f is always decreasing" with a table. If f(−2) = 4 and f(3) = −2, then for an input between them such as x = 0, the output a must lie between: −2 < a < 4.

Quick check

1. c(t) = cups of coffee, t = teachers. Which is dependent? c (cups of coffee) 2. f is decreasing on 1 ≤ x ≤ 5. What is true of f(2) and f(4)? f(2) > f(4)

3. What must happen to the graph at a point of inflection? Concavity switches

AP Precalculus Unit 1 · 1.1 2 of 2