Constant Function: f(x) = c

Absolute value function: f(x)=|x|

Square Root Parent Function: f(x) = √x

Quadratic Parent Function: f(x) = x^2

Graph:

Rational Parent Function: f(x)=1/x

Graph:

Graph:

Graph:

Graph:

**Parent Functions For Dummies**

Domain

The set of all possible x-values

Make the function "work", and will output real y-values.

Range

The set of all possible y-values

Depends on the input (domain)

Domains and Ranges

Domain: (-∞,∞)

Range: constant

y-intercept: y=c

x-intercept: none

Constant through (-∞,∞)

Even function

Symmetrical to y-axis

No asymptotes

Domain: (-∞,∞)

Range: (-∞,∞)

Y-intercept: y=0

X-intercept: x = 0

Increasing through (-∞,∞)

Odd function

Symmetrical to the origin

No asymptotes

Domain: (-∞,∞)

Range: [0,∞)

x-intercept: x=0

y-intercept: y=0

Decreasing from (-∞,0)

Increasing from (0,∞)

Even function

Symmetrical to y-axis

Asymptotes: none

Domain: [0,∞)

Range: [0,∞)

X-Intercept: X=0

Y-Intercept: Y=0

Increasing through [0,∞)

Neither even nor odd

No symmetry

No Asymptotes

Graph:

Domain: (-∞,0)U(0,∞)

Range:(-∞,0)U(0,∞)

No intercepts

Odd Function

Symmetrical to the Origin

Vertical Asymptotes: x=0

Horizontal Asymptotes: y=0

Decreasing: (-∞,0)U(0,∞)

Cubic Parent Function: F(x)=x^3

Graph:

Domain: (-∞,∞)

Range: [0,∞)

X-intercept: x=0

Y-intercept: y=0

Decreasing through (-∞,0)

Increasing through (0,∞)

Even function

Symmetrical to y-axis

No asymptotes

Domain: (-∞,∞)

Range: [0,∞)

X-intercept: x=0

Y-intercept: y=0

Increasing through (-∞,∞)

Odd Function

Symmetrical to the origin

No asymptotes

Greatest Integer Parent Function f(x)=[[x]]

How can you tell if a function is even or odd?

Graph:

Domain: All Real Numbers

Range: All Integers

X-intercept: x=[0,1)

Y-intercept: y= 0

Constant within each interval

Neither even nor odd

No symmetry

No asymptotes

If f(-x) = f(x) the function is even

If f(-x) = -f(x) the function is odd

**Pablo Is Awesome!!!!**

The points where the graph touches either the x-axis or y-axis

Make y=0 or x=0 and see the value of x or y at these points

What are asymptotes?

They are lines that the graph approaches as it gets closer to infinity.

They limit the domain/range of a function.

What are intercepts?

Exponential Growth Parent Function: f(x) = ab^(x-h) + k

Graph:

Domain: (-∞,∞)

Range: (k,∞) or (k,-∞)

Asymptote: y=k

Y-intercept:

y = a when h=0 & k=0

make x=o and solve for y when h & k are not 0

X-intercept: none

Increasing through (-∞,∞)

Neither even nor odd

No symmetry

Finding domain and range

Did you know?

In a function for every input there is exactly one output

Functions can be represented verbally, numerically, graphically, and algebraically.

Parent functions have a lot of children.

Increasing, decreasing, and constant intervals are related to the

monotone

of a function.

Functions can transform across the x-axis, up or down the y-axis.

If the graph is symmetrical to the y-axis, the function is even.

If the graph is symmetrical to the origin, the function is odd.

Transformations

There are two types of transformations:

Rigid

Horizontal shifts

Vertical shifts

Reflections: h(x) = -f(x), h(x) = f(-x)

Nonrigid

Change in shape

Stretches and shrinks

Vertical & Horizontal Shifts

c is positive real number

Vertical and horizontal shift in the graph of y=f(x)

Vertical shift

Upward: h(x)=f(x)+c

Downward: h(x)=f(x)-c

Horizontal shift

Right h(x)=f(x-c)

Left h(x)=f(x+c)

Tiffany Rocks!!!

Vertical and Horizontal Stretches and Shrinks

g(x) = cf(x)

if c > 1, vertical stretch

if 0 < c <1, vertical shrink

h(x) = f(cx)

if c >1, horizontal shrink

if 0 < c < 1, horizontal stretch