# Graphing Basics Who's Your Parent

## What are Parent Functions???

A parent function is the simplest function of a family of functions. You may be asking, what does this mean? In essence, it's just the bare bones, no frills representative of its family of functions (like linear, quadratic, cubic, etc.) and showcases the shape of its family of functions without any fancy numbers.

## So why are they so important in Algebra?

Parent functions can be used to figure out trends in a family of graphs, and are very helpful in graphing transform functions, functions who have been transformed in some way but are still part of their family, and thus still adhere to the same shape and rules.

## Understanding Transform Functions

Transform functions are basically the addition of variables to a parent function that tells us how to move or stretch the function (in essence, transform it) to look a certain way. Transform functions are super helpful in understanding to graph a function.

Key variables in transform functions:

a - This variable has two jobs. Its sign (whether positive or negative) determine which way the function faces. If this variable has a positive value, it faces the same way as its parent function. If this variable has a negative value, the function is reflected over the x axis. Additionally, this variable is responsible for the vertical stretch or compression of the function.

h - This variable determines the horizontal shift (left or right). If h is positive, the function is moved h units right; if h is negative, the function is moved |h| units left.

k - This determines the vertical shift (up or down). If k is positive, the function is moved k units up. If k is negative, the function is moved |k| units down.

## Common Function Families

This flyer will cover the some common function families, including:

Linear

Cubic

## Linear

Linear functions are, in essence, a sloped straight line.

They have one x intercept and one y intercept.

The parent function of a linear function is f(x) = x

This function goes through the origin. Both its y intercept and x intercept is (0, 0)

## Linear Transform Function

The linear transform function is a bit different from other functions. The function is as following:
f(x) = mx + b

In the above transform equation, m essentially takes on the role of the variable a in the other transform equations, and b takes on the role of variable k.

m determines the slope of the linear function - both how steep it is, and if it is reflected over the x axis.

b determines the vertical shift of the linear function. As with k, when b is a positive value, the linear function is shifted b units up; when b is a negative value, the function is shifted |b| units down. In the case of linear functions, b also represents the y value of the y intercept: ( 0, b)

Quadratic functions are also known as parabolas. Although the "steepness" of the function and its direction varies, it always forms a vaguely U shape.

The parent function of a quadratic function is f(x) = x^2

This function has its vertex at the origin. Both its x intercept and y intercept are at (0, 0). It has a step pattern of 1, 3, 5, 7 .... etc. that assists in plotting points for graphing based on an equation.

## Quadratic Transform Function

The quadratic transform function is as following:

f(x) = a (x - h)^2 + k

Refer back to the Understanding Transform Functions box on how to decipher the variables.

One thing to note about the variable a is to multiply it with each number in the step pattern to figure out how many y units lie between each increase in x units. For example, in the parent function, the step pattern to the positive infinity side of the x axis is right one, up one; right one, up three; right one, up five... etc. If you have a quadratic with the value 2 as the variable a, just multiply 2 by the step pattern of 1, 3, 5, 7.... to find the step pattern for the new equation. In this case the step pattern would become 2, 6, 10, 14, and so on.

## Cubic

Well, not much to say about it, except it looks like a scribble... (?!)

The parent function of a cubic function is f(x) = x^3. The x intercept and the y intercept, as well as a turning point, is located at (0, 0). It has a step pattern of 1, 7, 19, etc.

## Cubic Transform Function

The cubic transform function is as following:

f(x) = a (x - h)^3 + k

Refer back to the Understanding Transform Functions box on how to decipher the variables.

One thing to note about the variable a is to multiply it with each number in the step pattern to figure out how many y units lie between each increase in x units. For example, in the parent function, the step pattern to the positive infinity side of the x axis is right one, up one; right one, up 1; right one, up 7 ... etc. If you have a cubic with the value 2 as the variable a, just multiply 2 by the step pattern of 1, 7, 19.... to find the step pattern for the new equation. In this case the step pattern would become 2, 14, 38 and so on.

## Summary

Topics we covered today:

Parent functions

Graphing parent functions

Transform functions

Graphing transform functions

## Mark Your Calendars!

Quiz 4: Monday, December 7, 2015

Test 2: Thursday, December 10, 2015