precalculus & trignometry. objectives: recognize the graphs of parent functions. right now,...
TRANSCRIPT
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.