Sept. 27 Class

Learning Goals: Understand how to multiply and divide rational expressions.

Multiplying and dividing rational functions is much like multiplying and dividing fractions.  Make sure you always state the restrictions and be extra careful when dividing.  Here are notes I gave in class.

Homework: P. 121 #1–11

Sept. 26 Class

New Unit: Rational Functions

A rational function is any function that has a numerator and denominator, (ie. a fraction).

For example: .

We know that the denominator can never be zero, therefore we must state the restriction, x ≠ 0.

We can state the restrictions to any rational function by making sure the denominator does not equal zero.

Example: State the restrictions to .

Solution:  (x – 4)(3x + 2) ≠ 0, therefore, x ≠ 4 and x ≠ -2/3.


Rational expressions can be simplified if you have the same factor in the numerator and denominator.  Make sure to factor!

Example: Simplify and state the restrictions, .

Solutions: First we must factor to get, .  Now we see the same factor of x in the numerator and denominator.  Those "cancel out", leaving (2x + 1).  Note that even though the factor of x has been eliminated, the restriction still holds,

x ≠ 0

On a graph of this expression, you would find a hole at the restricted point.

It is very important that you state the restrictions from the original expression, before you simplify.

Example: Simplify and state the restrictions.

Solutions: First we must factor to get, . 

The restrictions are then x ≠ 3 and x ≠ -3.

Now we can simplify to get, .

Homework:  

Pg. 112  #1-11, 15

Pg. 107 #1eoo, 7, 8

Read, understand, and make notes: Pg. 116 (In summary)

Sept. 24 Class

Good luck in your preparations for the test tomorrow!

Here is the daily plan for the next unit: Unit 2 Daily Plan

Your primary job is to prepare for the summative test tomorrow.  If you are ready to move on, try the review to prepare for the next unit:

Pg. 88 #(4-6) eoo, 7, 8eoo, 11, 12

Pg. 95 #(5-6) eoo, 9, 10 11ac, 12

Pg. 102 #(1-7) eoo, 8, 9

Sept. 21 Class

Learning Goals: Review base functions and inverse functions.

Today we started by reviewing all the base functions and their domain and range.

Linear Function

Quadratic Function

Radical Function

Absolute Value Function

Reciprocal Function

 Cubic Function

 Exponential Function

You should also understand what happens to the domain and range when we transform these functions.


Next we reviewed the inverse of a function.

For a function, , the inverse is denoted .


Example: For the function f(x) = {(1, 2), (3, 2), (4, 5)}, what is the inverse?

Solution: The inverse is found by switching the x and y values,

Notice that this inverse is not a function, even though the original was a function!

The inverse of a function is not always a function.

Example: Find the inverse of f(x) = 3x – 5.

Solution: Follow these steps to find the inverse of an equation,

Example: Find the inverse of this graph.

Solution: You can take each point (x,y) and switch it, then plot the solution, 

OR 

you can reflect the graph over the diagonal line y = x.

Either way, you get this result:

Sept. 20 Class

Learning Goals:

• Understand how to transform a single point.
• Review function notation.

Today I discussed another way to think about transformations by looking at individual points instead of the entire graph.

Example: If the point (1, 1) is on the parent function of , what point is the image of the point on the transformed function ?

We can use a chart to organize our response.

Across the top row are the transformations listed in order.  On the second row we consider whether the transformation would change the x or y value.  (HINT: horizontal transformations change the x value, vertical transformations change the y value.)

Solution: The image of the point (1,1) is (-2, 7).

We can use a chart like this to figure out what would happen to ANY point (x,y).  To see this we can add another row to the chart:

Thus, we can see that any point (x, y) changes according to,

This is called the transformation map.  If you map enough points, you can use this method to graph the final function.

Review

At this point it is good to being reviewing what we know so far to prepare you for the summative text on Tuesday!

First of all, you must know what a function is.

Function: a relation in which every x value is related to ONE y value.

You must be able to identify functions and non-functions.  Remember to use the vertical line test if you have a graph.

You must be able to understand and use function notation: y = f(x)

Give a function, you must be able to evaluate it for different values of x.

Example: Given f(x) = 3x + 2 and g(x) = 5x.  Evaluate 2f(3) + g(4).

Solution: 2f(3) + g(4) = 2[3(3)+2] + [5(4)]
                                   = 2[11] + 20
                                   = 42

Example: Given f(x) = 3x + 2 and g(x) = 5x.  Evaluate f(g(x)).

Solution: This means that you substitute g(x) into f(x).

f(g(x)) = f(5x)
           = 3(5x) + 2
           = 15x + 2

The previous example is what we call a composite function.  This could also be written as the following,

You should be able to recognize that f(g(x)) and f og(x) mean the same thing.

Homework: 
Complete worksheet on Composite Functions.
Pg. 70 #16-21

Sept. 19 Class

Today we did a formative quiz!  Many of you missed it because of the photos, so here is the quiz.

Formative Quiz 2

After the quiz we took up the solutions,

Formative Quiz 2 Solutions

Unfortunately that took the whole class.  Where does the time go?

Homework: P. 70 # 1-13

Sept. 18 Class

Learning Goals: Understand how to combine multiple transformations and express them in function notation.

Today we discussed how multiple trasformations can be performed on any function.

The transformation of the function

f(x)

can be expressed as,

a f[k(x – d)] + c

|a| > 1, vertical stretch

|a| < 1, vertical compression

|k| > 1, horizontal compression by 1/|k|

|k| < 1, horizontal stretch by 1/|k|

-a , reflection in the x-axis (vertical)

-k , reflection in the y-axis (horizontal)

d > 0, horizontal translation to the right

d < 0, horizontal translation to the left

c > 0, vertical translation up

c < 0, vertical translation down

Order of Transformations: SRT

Make sure that you always do the transformations in the following order.

• Stretches first
• Reflections next
• Translations last

We then worked on this handout: Writing Transformations with Function Notation

Homework

P. 58 #1-8,10-12

Handout: Transformation of Basic Functions

Sept. 17 Class

Learning Goals: Understand how to vertical stretches affect the shape of a graph.

Today I handed back the formative quizzes.  Over the class did very well, good job!

Here are the solutions: Solutions to Formative Quiz 1

Then I took some time to discuss how vertical stretches effect the shape of a graph.  Vertical stretches are accomplished by multiplying a function by any value.  If the parent function is y = f(x), then the vertically stretched function is given by,

y = af(x)

All y values of the function are multiplied by the factor of a.

Here are the examples I worked out in class.  

You should now be able to complete the rest of the worksheets from last week.  If there are multiple transformations on a function, you must do the stretches first, then reflections then translations.

Homework Problems: Complete the rest of Functions Workshop and Vertical Stretches.

Sept. 14 Class

Learning Goals

• Understand how reflections transform a function.
• Understand how horizontal stretches transform a function.

Congratulations on completing your formative quizzes today!

After we quiz we discussed reflections and how they work.  If the parent function is y = f(x), then

y = -f(x) is a reflection in the x-axis.

y = f(-x) is a reflection in the y-axis.

I then worked out solutions to this worksheet:

Handout: Reflections 

Examples

A reflection in the x-axis.

A reflection in the y-axis.

Next we talked about horizontal stretches.  If the parent function is y = f(x), then the horizontally stretched function is given by,

y = f(kx)

By working out an explicit example we found that if the function is transformed with a k, the function stretches by a factor of 1/k.

Example

The original function is in black and the transformed function is in red.

In the above example, the value of k is 2.  Therefore the graph is "stretched" by a factor of 1/2.

(This can also be referred to as a compression by a factor of 2)

A reflection in the y-axis of the above example would look like this:

The reflected function is shown in blue.

Homework: All from the following handouts, 

Horizontal Stretches (Ignore vertical stretches pages.)

Functions Workshop (Parts I - VI)

Sept. 13 Class

Learning Goals: Understand how horizontal and vertical translations effect a function.

Today we discussed transformations.  The transformations that shift a graph left, right, up or down are called translations.  Translations don't change the shape of the graph, they just move the graph in one of the four directions.

We then worked through this handout that looked at how the domain and range change when you translate a function.

Translations of Functions

Here is a snippet of how you would fill out the table:

Then we discussed how the domain and range would change.
Here is another snap shot of what we did in class:

And finally, we did some examples of how to sketch transformations starting from the base graph.  In these images, red is the base graph and black is the transformed graph.

Remember, there is an asymptote in the graph of f(x) = 1/x.   When you translate the graph, the asymptote moves as well.  Normally we use a dotted line to denote the asymptote.

For homework please complete the rest of the handout.

Graphing tips
To check if you graphed correctly, you can use technology to check your answers.  Here are a few suggestions of free graphing apps that are useful:

• iPhone, iPod, iPad users, try downloading Quick Graph.
• Android users, try downloading Algeo Graphing Calculator.
• From any computer connected to the internet, you can try using this Online Graphing Calculator.

Sorry, I've never used a Blackberry before, but I'm sure it has it's own graphing calculator app.

You CANNOT use these apps on a test however.  Plus, I will always show you how to graph by hand.  These apps are just suggestions to help check your answers and to help you better understand the concepts.

Sept. 12 Class

Learning Goal: Understand how to find the inverse of a function.

Today we discussed what the inverse of a function means:

If a function takes an x as input and gives a y as output, then the inverse works in the opposite direction, taking the y as input and gives an x as output.

For an inverse the domain and range is reversed compared to the original function.

For a function, , the inverse is denoted .


Example: For the function f(x) = {(1, 2), (3, 2), (4, 5)}, what is the inverse?

Solution: The inverse is found by switching the x and y values,

Notice that this inverse is not a function, even though the original was a function!

The inverse of a function is not always a function.

Example: Find the inverse of f(x) = 3x – 5.

Solution: Follow these steps to find the inverse of an equation,

Example: Find the inverse of this graph.

Solution: You can take each point (x,y) and switch it, then plot the solution, 

OR 

you can reflect the graph over the diagonal line y = x.

Either way, you get this result:

There will be a formative quiz on the material so far this Friday.

Homework

Pg. 46 #1-7(eoo), 9-17