Math

Introduction

This transcript discusses the graphs of basic functions and their characteristics. Various types of functions are introduced, including linear, quadratic, cubic, and others, along with transformations of these graphs.

Linear Functions

The equation of a line in slope-intercept form can be expressed as:
$y = mx + b$

In function notation, this is written as: $f(x) = mx + b$
- Here, (m) represents the slope of the line, and (b) is the y-intercept.
If the slope (m) is not zero, the graph represents a slanted line.
If the slope is zero ((m = 0)), the equation simplifies to: $f(x) = b$
- This linear equation represents a horizontal line at the value of (b).
Such a function is categorized as a constant function since its output remains constant regardless of the input (x).

Basic Functions

Function Types and Their Graphs

Linear Function:
- Example:
  $f(x) = x$
- Slope: 1, y-intercept: (0,0)
- Visual representation: passes through the origin (0,0) and bisects the first and third quadrants at a 45-degree angle with the x-axis.
Quadratic Function:
- Example:
  $f(x) = x^2$
- Shape: Parabola (U-shape)
Cubic Function:
- Example:
  $f(x) = x^3$
- Characterizes the increase and decrease in output over changes in input.
Square Root Function:
- Example:
  $f(x) = ext{sqrt}(x)$
- Visualization: looks like one branch of a parabola rotated 90 degrees, starting at the origin.
Cube Root Function:
- Example:
  $f(x) = ext{cube-root}(x)$
- Graph reflects characteristics of the cubic function but with less steepness.
Absolute Value Function:
- Example:
  $f(x) = |x|$
- Shape: Forms a V with a point at the origin (0,0).
Reciprocal Function:
- Example:
  $f(x) = rac{1}{x}$
- The graph consists of two branches that approach but never touch the axes (the x-axis is a horizontal asymptote as (x) approaches infinity, and the y-axis is a vertical asymptote as (x) approaches 0).

Transformations of Functions

A transformation of a function can include vertical and horizontal shifts.
Vertical Shifts:
- For any function (f(x)) and a constant (k):
- $f(x) + k$ results in shifting the graph of (f(x)) up by (k) units.
- $f(x) - k$ results in shifting the graph of (f(x)) down by (k) units.
Horizontal Shifts:
- For any function (f(x)):
- $f(x + h)$ translates the graph left by (h) units.
- $f(x - h)$ translates the graph right by (h) units.
- The direction shifts opposite to the sign of (h).

Examples of Function Transformations

Parent Function:
- Example:
  $f(x) = |x|$ (absolute value function)
- Transformed function:
  $g(x) = |x| + 1$
- Expectation: Graph should resemble (f(x) = |x|) shifted vertically up by 1 unit.
Horizontal Transformation:
- Example:
  $g(x) = ext{sqrt}(x + 5)$
- Expectation: The graph is the same as (f(x) = ext{sqrt}(x)), shifted left by 5 units.
- Domain changes: ( f: x \geq 0 ) becomes ( g: x \geq -5 ).
Example X-intercepts:
- For (y = ext{sqrt}(x)), the x-intercept is (0,0).
- For (g(x) = ext{sqrt}(x + 5)):
  - Setting equation to zero:
    $0 = ext{sqrt}(x + 5)$
  - Solve:
    $x + 5 = 0 o x = -5$
  - X-intercept: (-5,0)

Rigid Transformations

A transformation that moves the entire graph without altering its shape is termed a rigid transformation.
Examples include moving the graph of the parent function (y = x^2) with strategic points and applying transformations separately, confirming the shape remains the same (e.g., paralleling a wire that shifts with motion).

Piecewise Functions

Definition: A piecewise function consists of multiple parts, each defined by a different expression based on its domain.
Visualizing piecewise functions can utilize graphing calculators.
For example, if entering into a graphing utility:
- $y_1$ for the first piece:
- $y_1 = -2.7x - 4.1$ for x <= -1
- The function can have separate rules for other intervals as well (e.g., $y_2 = -x^3 + 2x + 5$ for -1 < x < 2).

Evaluating Functions

Evaluation of a given function at specific x values:
- For example, given the function defined by multiple equations, we seek values like:
- $g(-2) = ext{use appropriate rule depending on interval}$ .
- Demonstrating how x-values determine which function piece applies provides clarity.

Even and Odd Functions

For a function (k(x)):
- Find (k(-x)) to determine if the function is odd or even:
  - If (k(-x) = k(x)), then the function is even.
  - If (k(-x) = -k(x)), then the function is odd.
- For example,
  $k(x) = 13x^3 + 12x$
- Evaluate:
  $k(-x) = 13(-x)^3 + 12(-x) = -13x^3 - 12x$
- Since ( k(-x) \neq k(x) ) and ( k(-x) = -k(x) ), it indicates that (k(x)) is an odd function.