Fitting the Pieces Together for TAKS

Algebra Objectives 1 – 4

The first 4 objectives revolve around what a function is, what features it has, with particular emphasis on linear functions.

In today’s session, we are going to cover the topics of:

• Independent & Dependent Variables• Finding Patterns• Domain & Range• Interpreting Graphs• Parent Functions• Parameter Changes

The next time we cover these same objectives, we will cover:

• Simplifying Expressions• Setting up & Solving Equations• Multiple Representations of Functions• Slope –y-intercept form• Point-slope form• Direct variation• Setting up & Solving Inequalities• Setting up & Solving Linear Systems

One thing that has consistently been tested is your ability to look for patterns.

Algebra is the study of patterns.We wouldn’t have formulas if it wasn’t for patterns.There are numeric patterns and there are geometric

patterns.Some patterns are linear, some are quadratic, some are

exponential, and yet others represent other functions.

Some patterns don’t even represent functions.

Here is a sample of how patterns have been tested on TAKS.

You would be expected to study the table, look for patterns in the data, and extend what you see here to make a prediction about something that is not in the table.

Study the table and tell me what you see.

The dates are increasing at 5 year intervals.

The values in both columns are increasing.

The number of passengers are increasing by 7.9 million every 5 years.

Continuing the pattern, we could add a couple more rows.

20052005 70.170.1

20102010 7878

Now, remember, we were asked about 2008, which is in between the two years that we added onto the table.

Here are the options we were given.

Notes that you might want to make for yourself:

• Study the table• Look for patterns: adding & subtracting a

constant value will indicate a linear pattern• Extend the pattern• Use the options!!!! Eliminate those options

that don’t make sense• Be sure to underline what question the

problem is asking.

Sometimes, you will be

asked to come up with a

function rule for the

pattern.

Study that pattern!

The x-coordinates are increasing by 1 each time.

However, those y-coordinates are not increasing by a constant value.

This pattern is NOT LINEAR!

Look at the options.

Options F & G are both linear, so they CANNOT be the answer.

How do I know they are linear? The equation has no power higher than the understood 1 for either x or y.

Make use of that calculator!!!!

Now, type those other two equations in y1 and check out the table!

You want the rule whose table on the calculator matches the table on the test.

NOPEYEP!

If all else fails, you can use the calculator and check all 4 options to see if the tables match!

Notes you might want to make:

• Anytime function rules and tables are used in the same problem, use the calculator!

• Enter function rules in y= and see if the tables match.

Remember that if there is a dependence of one item upon another, then there exists an

independent and a dependent variable.

How much I get paid depends upon how many hours I work.

How much I get paid is dependent

How many hours I work is independent

How quickly I get from my house to school depends on how fast I am driving.My time getting from home to school is dependent.My speed is independent.

How long a song is determines how many songs I can put on my ipod.

The length of a song is independent.

The number of songs on my ipod is dependent.

TAKS asks questions like this:

If you were the vendor, how would you label your y-column? Temperature? Or # ice cream cones sold?

What quantity depends on the other?

TAKS asks questions like this:Since the number of ice cream cones sold depends on the temperature, the dependent quantity is number of ice cream cones.

Remember, MIXID and DRYORM manipulated variable (science term)I independent variable (math term)X x-coordinatesI input (numbers you put into the rule)D Domain (set of all x-coordinates)D Dependent variable (math term)R Responding variable (science term)Y y-coordinatesO output (answers you get from rule)R Range (set of all y-coordinates)

Knowing MIXID and DRYOR helps with Domain & Range.

Domain is the set of all of the x-coordinates that make up a graph or could be substituted into a rule/equation.

Range is the set of all the y-coordinates that make up a graph or could be the answers from a rule or equation.

Try this:

We need to identify the DOMAIN.

Domain is the set of all the x-coordinates, so follow the x-axis (it’s labeled).

The x-coordinates start

here at -3

And the graph continues to

have points as we move right

Until it stops where the x

is 6.But notice that the point where x is 6 is open! The x-coordinates go up to, but do not include 6.

The x-coordinates started at -5 and went up to 6. -5 is included because the point is filled. 6 is not included because its point is hollow.

Here were the answer choices

The bar/equal means included, as with -5. If it’s missing, that number is not included, as with 6.

x is between -5 and 6, just like the red line above.

You might also consider drawing a frame around the graph.

For the domain, you would follow the frame from the left end to the right end.

For the range, you would follow the bottom to the top.

DOMAIN

RANGE

Within these 4 objectives, you will also need to interpret graphs.

The test will provide you with 4 graphs to choose from. You want to reread the problem and make sure that you know what it says.

You want to pay attention to the independent & dependent variables.

You want to make sure that you understand what each graph is telling you.

Here is the first graph.

What is the independent variable?

What is dependent?This graph says that Karen

started from a still position. As time went on, her speed increased at a steady pace at first and then slowed down at a steady pace.

Does this interpretation match the situation described?

No

Time

Speed

Let’s look at the next graph.

This graph says that Karen started from a standstill, increased her speed over time at a steady pace, and then maintained that maximum speed. How does this

interpretation fit with the situation?

It doesn’t.

What about the 3rd graph?

Karen starts at her maximum speed and maintains it for a while. Then, she slows down steadily.

This graph doesn’t fit the situation, either.

Looks like it has to be the final graph. Interpret it just to be sure. DO NOT

ASSUME!Karen starts at her a

medium speed and maintains it for a while. Then, her speed increases.

This graph seems to fit the situation: Karen jogs at a steady pace in the neighborhood. Then, runs downhill and her speed increases.

Sometimes, you have to interpret the graphs more specifically.

What is independent? What is dependent?

Concentrate on the question asked.

Distance depends on Force 10 in.

requires 50 lbs of

Force

4 in. requires 20 lbs of

Force

The difference between the forces means subtraction.

50 lbs – 20 lbs = 30 lbs

Look at the options.

For F, they subtracted the

distances.

For G, they added the distances.

For J, they added the forces.

Let’s talk parent functions, now.

The TAKS test covers only the functions you studied in Algebra 1.

• Linear functions• Quadratic functions

However, you know more than those two functions, so you might see the other ones used on the test.

Parent functions are the most basic, no extra frills added equations—the bare minimum necessary to get the

required graph.

• linear: y = x• quadratic: y = x2

You are expected to know what the equations and the graphs look like.

Linear

Quadratic

The absolute value function

The square root function

See, they used the other functions that you studied, but they were NOT the answer!

And you are supposed to be able to tell what changes in the equation

make specific changes in the graphs.

Translations, Reflections, Dilations

Study the graph first!

They do NOT want the equation of THIS line. I guarantee its equation will be one of the choices!

We want the line that is translated DOWN 2 units.

That will give a y-intercept of 1

Look at the options. Eliminate the ones that do not have 1 as the y-intercept. (You

know, y=mx+b and b is the y-intercept!)

The only difference between the two remaining options is the slope! Go back to the graph and count the slope.

The translated line just went down 2

units. It didn’t change steepness.

Just count the slope on the given line, since our line

must have the same slope.

Start at the y-intercept. The first option had a slope of 3. That means up 3 and right 1.

Make that movement. Are you on the line?

NO

Okay, that leaves option G.Check; DON’T ASSUME.

Option G has a slope of 1/3. That means up 1 and right 3, starting at the y-intercept.

Make the movement.Are you at the line?

YES

The answer is G.

That was a lot for today!

And there is still a lot more to do.

Right now, let’s practice on some other problems like the ones we just did.

2

2

3

3

3

3

y x

x y

x y

y x

A.

B.

C.

D.

Your answer is D

A.

B.

C.

D.

3 3

3 3

5 4

5 4

x

x

x

x

And the answer is B

D is theanswer

A. Its value at 18 months was twice its value at 36 monthesB. It value at 36 months was half its value at 54 monthsC. It depreciated $200 every 12 monthsD. It depreciated $400 every 18 months.

A. The new line is parallel to the originalB. The new line has a greaer rate of changeC. The x intercept increasesD. The y intercept decreses.

The answer is C

A. ExponentialB. Absolute ValueC. LinearD. Quadratic

The answer is C

A. The graph has its vertex at the originB. The graph is a parabola that opens upwardC. The graph has the x axis as its line of symmetryD. The graph as a minimum value at (0,0)

The answer is C

4

3

4

3

A.

B. V

B. R

C.

The answer is B