A LIBRARY OF PARENT FUNCTIONS

Precalculus & Trignometry

Objectives:

Recognize the graphs of parent functions. Right now, you are responsible for

Linear functions Quadratic functions (squaring function) The square root function The cubic function The cube root function Reciprocal functions Absolute Value function

Graph all of the above functions Differentiate between even and odd functions

Even functions

Symmetric with respect to the y-axis

Odd functions

Symmetric with respect to the origin

Think of an odd function as each point being

Rotated 180

Linear Functions

Equation: The graph is a line that has a slope

of 1 and passes through the origin Also called the identity function

(because the x and y values are the same.

D Range: The function is odd

Graph of the parent linear function

Plot and then extend theLine by using a slope of 1 inEach direction.

Quadratic Functions

Equation: Graphis called a parabola Its vertex is at the origin and is a U-

shaped curve Domain: Range: Function is even

Graph of the parent quadratic function

Plot

Square Root Function

Equation: or Domain is Range is The graph is neither even nor odd

The parent square root function

Plot

X values that are not perfectSquares will lead toIrrational y values which areVery difficult to plot

Cubic Function

Equation: Domain: Range: Odd Function

Graph of the Cubic Parent Function

Plot

Cube Root Function

Equation: or Domain: Range: Function is odd

Graph of the Cube Root Function

Plot

Using x values that are notPerfect cubes will result in Irrational y values that areVery difficult to plot.

The Reciprocal Function

Equation: Graph is called a hyperbola Domain: or Range: or Graph has a vertical asymptote at

and a horizontal asymptote at so the graph will approach positive and negative infinity.

The function is odd.

Graph of the Reciprocal Function

Do not allow the graph to touch orCross the x-axis or the y-axis.

Plot (1, 1) and draw a curve

Plot (-1, -1) and draw a curve.

Absolute Value Function

Equation: Domain: Range: Actually a piecewise function made

up of Even function

Graph of the Absolute Value Function

Plot (0, 0) and then graph aSlope of positive 1 to the rightAnd a slope of negative 1 toThe left to create a V shape.

Practice Graphing

You will be given an equation like , etc and be expected to graphthe function quickly, without using a calculator or making a table. You should practice this.

You will also be given a graph and expectedto write the parent function equation.

You will also need to know the domain and range of each function.

Lastly, you will need to know which functions are even and odd.

Transformations of Parent Functions

Shift functions vertically (up and down)

Shift functions horizontally (left and right)

Stretch and compress functions vertically

Reflect functions over the x and y axis

Vertical & Horizontal Shiftsjust moves the graphdoes not change shape

Shifts the graph “c” units up

Shift the graph “c”’ units down

Shifts the graph “c” units to the RIGHT

Shifts the graph “c” units to the LEFT

Example from Desmos

Another Example from Desmos

Vertical Stretches and Compressions

, k > 1 Vertically stretches the graph by “k” Keep the “x” values the same but

multiply the “y” values by “k.” The graph will get skinnier from left to

right the higher “k” is.

Vertical compression The graph will look wider from left to

right Keep the “x” values the same and

multiply the “y” values by k.

Reflections

Functions can be reflected over the x-axis or the y-axis.

reflects the graph over the x-axis. reflects the graph over the y-axis

Some graphs like the quadratic and absolute value function remained unchanged when reflected over the y-axis. Why?

Desmos Example

What you will need to be able to do: At this point, you should be able to

graph al of the parent functions. You should then be able to look at an

equation of a function, know which parent function it represents, and then apply the correct transformations to graph the new function.

Example:

You should be able to ignore the “-3” and “+4” and realize that this is just a parabola; however, the parent function has been shifted to the right 3 and up 4 units.

To graph it, you just shift each original point from the parent function right 3 and up 4.

Graphs of the example

I I labeled the originalPoints A, B, C, D, E andTheir transformed pointsAs

Those are read as “A prime”“B prime” etc.

Example:

The parent function is just The transformations are a flip over

the x-axis (from the negative in front), a vertical compression by ½ (from the ½ in front) and a horizontal shift to the left by 1 (from the 1 inside the parenthesis)

Graph the transformation by moving each point on the parent function accordingly.

Graphs

A Gift for Making it Through this Presentation Here’s a chance for bonus. The first

20 emails that I receive with the correct answer will be awarded the bonus points.

You must explain your work in the email to receive credit.

You must also provide me with a written copy of your work when we return to school, or you will lose the bonus points.

Go to the next slide for the problem.

A Rose Garden

Suppose that you are working for the mayor of Pittsburgh. Your job is to complete a project that would add a border of yellow roses around an existing rectangular rose garden. The current rose garden is 12 ft long and 5 ft wide and contains only red roses. The yellow rose border should be the same width (thickness) on all 4 sides and should have the same area as the current red rose garden (because the planning committee bought the same number of red and yellow roses to begin with, and we don’t want to waste roses!)

Determine the width of the yellow rose border to the nearest tenth of a foot.

A little help

?

I want the width/thickness of the yellow rose border. That is the same on all 4 sides.It’s the distance from the outer edge of the red roses to the outer edge of the yellow Rose border. I marked it with a green line for you.