AP Precalc 1.1 Change in Tandem Notes

1.1 Change in Tandem

Function

• Definition: A function is a mathematical relation that maps each input value to exactly one output value.
• Input Values:
  • Also known as the Domain.
  • Represented by the Independent variable (often "x").
• Output Values:
  • Also known as the Range.
  • Represented by the Dependent variable (often "y" or $f(x)$).
• Variable Representation: It's useful to use variables that represent what you are measuring. For instance, when throwing a football, $H$ is the height, and $t$ is the time in seconds.
• Function Notation: Helps to identify input and output variables.
  • Example: $H(t)$, where $t$ is the input (time) and $H(t)$ is the output (height at time t).

Example

• $w(x)$ is a function.
  • $w$: amount of water in a pool (gallons).
  • $x$: length of the pool.
  • Independent variable: $x$ (length of the pool).
  • Dependent variable: $w(x)$ (amount of water in the pool).

VANG: Ways to Represent a Function

The rule that relates the variables in a function can be expressed in four ways:

• Verbally
• Analytically
• Numerically
• Graphically

Increasing Function

• Verbally: As input values increase, the output values always increase.
• Analytically: For all $a$ and $b$ in the interval, if a < b, then f(a) < f(b).
• Graphically: A graph that rises from left to right.
  • If a < b, then f(a) < f(b).

Decreasing Function

• Verbally: As input values increase, the output values always decrease.
• Analytically: For all $a$ and $b$ in the interval, if $a < b$, then $f(a) > f(b)$.
• Graphically: A graph that falls from left to right.
  • If $a < b$, then $f(a) > f(b)$.

Numerical Example of Increasing/Decreasing Function

• Determine if the function is increasing or decreasing on the given interval.
• Example:
  • Given a table of $x$ and $f(x)$ values, check if the function is increasing/decreasing.
  • If for all $a$ and $b$ in the interval, if a < b, then f(a) < f(b), the function is increasing.

Basic Elements of a Function's Graph

• Zero:
  • The graph intersects the x-axis (independent axis).
  • The output value is zero at this point.
  • Input values are called Zeros.
• Y-intercept:
  • The graph intersects the y-axis (dependent axis).
  • The input value is zero.
  • The output value is the y-intercept.
• Concavity:
  • Concave Up: Bowl facing up; Slope INCREASING.
  • Concave Down: Bowl facing down; Slope DECREASING.
  • Straight lines don't have concavity.
• Point of Inflection:
  • The point where the concavity changes.
  • Where the slope is the steepest/changing the fastest.

Graph Analysis Example

• Given a graph $f(x)$, determine:
  • When the graph is concave up.
    • Example: $(-\infty, i) \cup (K, \infty)$
  • When the graph is concave down.
    • Example: $(i, k)$
  • Find the zero(s) of the function.
    • Example: $x = g, x = m$
  • Find the y-intercept(s) of the function.
    • Example: $y = a$
  • When the graph is increasing.
    • Example: $(h, j) \cup (l, \infty)$
  • When the graph is decreasing.
    • Example: $(-\infty, h) \cup (j, l)$

1. 1 Practice

Identifying Independent and Dependent Variables

• Example 1:
  • $b(s)$ is a function where $b$ is the number of books in the library, and $s$ is the number of students in the school.
    • Dependent: number of books in the library.
    • Independent: number of students in the school.
• Example 2:
  • $p(t)$ is a function where $t$ is the number of years since kindergarten, and $p$ is the number of Pokemon cards.
    • Dependent: number of Pokemon Cards.
    • Independent: number of years since Kindergarten.
• Example 3:
  • $c(t)$ is a function where $c$ is the number of cups of coffee consumed, and $t$ is the number of teachers at the school.
    • Dependent: Number of cups of coffee consumed.
    • Independent: Number of teachers at the School.
• Example 4:
  • Area of a circle as a function of its radius.
    • Dependent: area of a circle.
    • Independent: radius of a circle.
• Example 5:
  • Volume of a box as a function of its side length.
    • Dependent: volume of a box.
    • Independent: side length of a box.

Determining Increasing or Decreasing Intervals

• Given a function $f(x)$, determine if it is increasing or decreasing on a given interval.
• Example Scenarios (Numerical):
  • Provide a table of $x$ and $f(x)$ values.
  • Determine if the function is increasing/decreasing based on the values.
  • If for all $a$ and $b$ in the interval, if $a < b$, then $f(a) > f(b)$, the function is decreasing.
  • If for all $a$ and $b$ in the interval, if a < b, then f(a) < f(b), the function is increasing.
• Example Scenarios (Graphical):
  • Given a graph of $f(x)$, determine the interval(s) where it is increasing or decreasing.
  • Increasing: as x increases, y increases
  • Decreasing: as x increases, y decreases
• Using inequalities and interval notation to describe intervals

Graph Analysis Practice

• Given a graph $f(x)$, determine:
  • Interval(s) where the graph is concave up.
    • Express answer using inequalities and interval notation.
  • Interval(s) where the graph is concave down.
    • Express answer using inequalities and interval notation.
  • Interval(s) where the graph is increasing.
    • Express answer using inequalities and interval notation.
  • Interval(s) where the graph is decreasing.
    • Express answer using inequalities and interval notation.
  • Zeros of the function.
  • The y-intercept of the function.

1. 1 Test Prep

Multiple Choice Questions

• Question 16:
  • Given a decreasing function $f$, and a table of selected values of $x$ and $f(x)$, determine all possible values for a variable $a$.
• Questions 17-18: Use a given graph of a function $f$ to answer the following questions. Let $a, b, c, d$, and $e$ represent the x-coordinates at those points.
  • Identify the point of inflection.
  • Determine on which interval $f$ is decreasing and the graph of $f$ is concave down.