Math Models & Algebra I CBA Review

Tips and tricks to taking the test…

-Read the question THOROUGHLY and highlight what they are actually asking

-look for keywords and terms (such as difference, range, domain and rate)

-remember NOT to rush through questions

-remember to read what the x and y axes are when looking at a graph

-read ALL the answers and compare to the question being asked.***Sometimes all answers are correct- but which one is the correct answer for what is being asked?

Common Representations of Slope and equations

-Slope: the rate of change, the steepness on a graph

Slope can be negative or positive,….

Or slope can be constant (zero) or Undefined…

REMEMBER: A vertical line is UNDEFINED; someone is trying to divide the slope by ZERO when given 2 points

Intercepts are when the line crosses the x or y axes

-An x-intercept is when the line crosses the x-axis (AKA a ZERO)-when given a table it is when the y value = 0

-A y-intercept is when the line crosses the y-axis -when given a table it is when the x value = 0

Slope-Intercept Equation of a Line

The form y = mx + b is called slope-intercept form because It gives the user both the slope of a line (m) and the y-intercept (b)

y = mx + b

y = 2x + 3

m = 2, and is the slope of the line

b = 3, and is the value of y when it hits the y-axis

***The y-intercept is ALWAYS inThe form (0, b)***

Manipulating equations and graphs – AKA parameter changes

When give an equation in the form y = mx + b, the graph can be manipulated by making changes to m or b.

If we increase the size of m (POSITIVE OR NEGATIVE), we make the slope steeper and ROTATE the lineabout the y-axis

If we decrease the size of m (think decimals & fractions)we make the slope LESS steep and ROTATE the lineabout the y-axis

Comparisons-Ratios and Proportions

Sometimes we need to compare two numerical values

For example, look at the slope of 2

When written as a fraction, it is 2/1 (spoken as 2 to 1)-This means we move up 2 and over 1 on the graph

Sometimes we can compare 2 types of data that are proportionalAnd guess what value will come next.

EX 1: In this case, when x Increases, the value of yIncreases by a value of 3.This is a 1 to 3 proportion

Too much math, too soon… Have a laugh!

The red arrowShows that theTeacher wasNOT amused

Review Problems

What is the equation of this line?

What are we given?The graph shows the y-interceptAs (0,-3)

The points give to us are (-1, -5) and (4, 5)

We can calculate the slope with

The equation

The value of the slope of this line is m = 2

We now know that theSlope is m = 2 and the Y-intercept is (0,3)

The equation of a line Is the form y = mx + b

To find the equation of thisParticular line, we just Substitute the values of m and bWith what we found

y = mx + by = 2 x - 3

But what if they ask for somethingCrazy like-

What happens if we increase the Slope to 4?

THINK BACK TO WHAT YOU KNOW

A larger m value means a Larger slope which means…

STEEPER LINE!

You can recognize it by figuring out the equation of the line andMultiplying the m value by 4, OR you can look at graphs and seeImagine the line ROTATING counter-clockwise to become steeper

GRAPHSSSSSSSSAm

ount

of t

ime

Wat

chin

g Sp

onge

bob

Time periods while in school

Not kidding. This may be an amusing graph, but at the same timeIt represents a lot of things.

Amou

nt o

f tim

eW

atch

ing

Spon

gebo

b

Time periods while in school

This graph shows trends- it shows what time periods areIncreasing or decreasing.

Analyzing the graph, we see that we don’t watch as muchSBSP between middle school and high school. The amount doesn’tIncrease or decrease. It stays the SAME (0 slope).

It decreases after elementary and INCREASES after high school

You can wish for a “rain day”, but it’ll probably not happenIn San Antonio any time soon ;)

We know how to get slope from an equationIn the form y = mx + b

Too easy! It’s always the number ATTACHED to the x.

But what if it ISN’T in slope-intercept form?

Example:

2y = 12x + 4

It LOOKS like y = mx + b except one MAJOR distractor

y MUST be by itself.

THEY ARE TRYING TO TRICK YOU!

We want the equation in the form y = mx + b-which means y MUST be by itself!

m = 6 and b = 2

If we wanted to find theY-intercept, we HAVE toBe careful during multiple choice

They could put (0,2) and (2,0)

You MUST remember that the y-intercept ALWAYS ALWAYS SIEMPRE ALWAYS has x = 0.

Elisa made horchata for her Abuela’s birthday party. The graphModels the relationship between the number of glasses of horchataServed and the amount of horchata remaining.

(Because, you know, when getting ready for someone’s birthday youHave all this extra time to sit around and make graphs…)

In this situation, what does the x-intercept represent?

a) Elisa is serving 12 ounces of horchata per glassb) Elisa started with 320 ounces of horchatac) Elisa has enough horchata for about 26 servingsd) Elisa can serve about 5 people per gallon

a) Elisa is serving 12 ounces of horchata per glass-this is TRUE but what does it have to do with (26, 0)?

b) Elisa started with 320 ounces of horchata-this is also TRUE but what about (26, 0)?

c) Elisa has enough horchata for about 26 servings-this IS the point (26, 0) directly affect x-intercept

d) Elisa can serve about 5 people per gallon-this is FALSE but does it have anything to do with (26, 0)? NO!

This question is asking forThe x-intercept. NOT the slope,Not starting amount. The x-intercept ONLY.

On a certain day the exchange rate of Mexican pesos for U.S. dollars was approximately 10 pesos for 1 dollar. If an exchange of 4,000 pesos was made that day, what was the approximate value of the exchange in dollars?

a) $40b) $400c) $4,000d) $40,000

Think back to rates and proportions.

10 is to $1 as 4,000 pesos is to …?

10 pesos for every 1 dollarIf we had 4,000 pesos, would we divide or multiply by 10?

Knowing 10 pesos = $1 shows that the dollar has a higher value The more pesos you have, the less dollars you have In this case we would divide 4,000 pesos by 10 Finally we have $400 for 4,000 pesos

The following graph shows the rate at which gas is leavingThe gas tank depending on the type of car that is drivenAnd how far it is driven. For this graph- Which car is in 2nd place for best gas mileage?

Look at ALL the lines and THINK about what is happening.

All slopes are DECREASING. But does mean any of these lines are “slow”?

NO! Some are slower or faster than other.

We want to find the BEST gas mileage- what does that mean?

We want to find the car that Goes the FARTHEST on one tank of gas.

For this graph, we look for the Highest x-value.

What other assumptions canBe made from this graph?

Maserati GranSport 2012 Concept Art

For a car traveling at a speed of 150 miles per hour, the relationshipBetween the distance traveled, d, and the time traveled, t, isDescribed by the function d = 50t. Which of the following Statements is true?

a) The time traveled depends on the distance traveledb) The distance traveled depends on the time traveledc) The speed of the car depends on the distance traveledd) The speed of the car depends on the time traveled

For a car traveling at a speed of 150 miles per hour, the relationshipBetween the distance traveled, d, and the time traveled, t, isDescribed by the function d = 50t. Which of the following Statements is true?

Let’s look at what is going on. The function we are asked to evaluateIs d = 50t.

This is in the form of y = mx + b.

We know that we is always DEPENDENT on x which is INDEPENDENT.

Here, if we line up the two equations we see…

Now let’s look back at our choices.

For a car traveling at a speed of 150 miles per hour, the relationshipBetween the distance traveled, d, and the time traveled, t, isDescribed by the function d = 50t. Which of the following Statements is true?

a) The time traveled depends on the distance traveledb) The distance traveled depends on the time traveledc) The speed of the car depends on the distance traveledd) The speed of the car depends on the time traveled

Distance is y and DEPENDENTTime is x and INDEPENDENT

Therefore, distance DEPENDS on time

No, whale is NOT an answer on your exam…(Dr. Todd’s kids know the answer is ALWAYS unicorn!)

Function Notation Tutorial

y = mx + bf(x) = mx + b

Guess what? f(x) is the same as y.It is the shorthand for “y is a FUNCTION of x”

What if we have f(x) = 2x + 1, and x = 5?

Then we substitute in: f(5) = 2x + 1 f(5) = 2(5) + 1

f(5) = 10 + 1 f(5) = 11

But what happens when theyGive us f(2) = 6?

OH NO NO NO NO NO…Wait! It’s ok.

We know f(2) = 6

So if we are asked to find x, it’s OK…

Don’t freak out. DON’T be a Squidward.

If f(x) = x + 4,What is x if f(x) = 6?

No worry! We know y is a function of x.y = 6. Just substitute and solve.

6 = x + 4-4 -4 2 = x

Let’s make it harder.

Which function includes the following data set?

(2, 4 ) and (9, 18 )?

a) Y = x/2 b) Y = 2x - 9c) Y = 2xd) Y = (x/2) + 3

A point is made up of TWO parts- an x and a y.

Plug x into each of the equations and see what your y is.