university prep math

University Prep MathPatricia van Donkelaar

[email protected]://pvandonkelaar.hrsbteachers.ednet.ns.ca

Course Outcomes

By the end of the course, students should be able to1)Solve linear equations2)Solve a system of equations3)Identify linear, quadratic and exponential patterns 4)Algebraically find the equation of each type of pattern5)Graph linear and quadratic functions6)Transfer between the 3 forms of a quadratic7)Solve quadratic equations by factoring, completing the square or by using the

quadratic root formula8)Solve for the vertex of a quadratic9)Solve exponential equations using common bases and logs10)Solve logarithmic equations11)Solve simple probability problems12)Use permutations and combinations to solve problems involving probability

Definition Example

variable a letter used in the place of a variable value

The ‘x’ in 2x – 6 is a variable. Its value could be 4, or –6, or any other number.

expressiona mathematical phrase which contains numbers, variables, and/or operators

2x – 6 is an expression. We can’t and don’t know anything about the value of the variable, x.

evaluating an expression

determining the value of the expression once the value of the variable(s) is(are) given.

Evaluate: 4x – 1 when x = 3.We plug-in a 3 wherever we see an ‘x’ and follow the order of operations.

equation two equivalent expressions on either side of an equal sign.

2x – 6 = 4 is a simple equation. We can solve to find the value of x.

solving an equation

orfinding the root(s)

determining the value(s) of a variable when the value of the expression is given. This/these value(s) is/are called solution(s), or root(s)

Solving the equation2x – 6 = 4 would give a solution (root) of x = 10.Solving the equation x2 = 4 gives roots 2 and –2

Order of Operations: When to do what

RACKETS

XPONENTS

IVISION

ULTIPLACTION

DDITION

UBTRACTION

B E D M A S

Evaluate the following expressions given the value of the variable stated.

1) 7x – 3 if x = 7

2) 10(x – 2) if x = 4

3) 5r – 7t –6 if r = 2 and t = 1

4) 3t2 +5t – 9if t = 2

5) if x = 4

6) if j =3

Answers:1) 462) 203) –34) 135) 266) 1

2

23 xxx

jj

2

5

Find the root(s) of each equation.

1) 5(x – 4) = 10

2) 8w – 2 = –42

3)

4) 3x + 6 = 9x – 4

5)

6) 7m – 4 = 2m – 19

7) x2 + 1 = 26

913

4

r

Answers: 1) 62) –5 3) 114) 10/6 (or 1.666…)5) 266) –37) 5 and –5

6121

g

FunctionsA function is a relationship between two variables(where each permissible value of the independent variable corresponds to only one

value of the dependent variable)

y = x2 + 3 is a relationship between two variables(x and y). In this case, the function says“y is always 3 more than x times itself”.

p = 1.23f is also a function.It shows the relationshipbetween f (liters of fuel)and p (price)

The simplest functions are linear

liner

not linear (quadratic)

Linear FunctionsThe temperature in Dallas, Texas is 94°F. What is that

temperature in degrees Celsius?

…but if we had the mathematical formula for the relationship (EQUATION OF THE LINE) we could find the answer exactly.

Here is the graph of the (linear) relationship between Celsius and Fahrenheit.

We can use it to estimate the answer….

3259

CF

The equation

y = mx + bdescribes any straight line!

For a specific line, m (slope) and b (y-intercept) are fixed or constant values, and represent actual slope value and the actual y-intercept value for that specific line.

The x and y are variables that have the certain relationship as determined by the equation, so they stay in the equation.

THE EQUATION OF A LINE

The SLOPE (m) of a line is a number indicating its steepness.

Each of these lines have a different slope,but the same y-intercept, b = 3.


m = 1

m = 3

m = −1/2

m = 0

THE EQUATION OF A LINEThe y-INTERCEPT (b) of a line is the point where the line crosses the y (vertical) axis.

Each of these lines have a different y-intercept,but the same slope m = –2.

b = −1b = −5 b = 3 b =5

You can think of them together as the PIN of the line. Once they are know, then we have full access to all the line’s information, and can use it to solve problems.

We can:• draw and use the graph of the line• find and use points on the line• write and use the equation of the line• solve problems using the linear relationship• etc…

Each distinct line has a specific slope (m) and a specific y-intercept (b).


But how do we find thesetwo very important values?

• If we have the equation, it’s easy-peasy (if the equation is y = 3x + 5, then m = 3 and b = 5)

• If we have the graph, b is usually easy-peasy (just find the y-value where the line crosses the y-axis), but m might take some work (see next slide)

• If we have at least 2 points on the line, calculate m first (see next slide), then calculate b (three slides from now)

• In a word problem, the rate is m, and the initial value of the y-variable is b.


Finding slope:


The slope is a measure of how muchchange there was for y (the dependentvariable) for every change in x (theindependent variable).

Mathematically we divide the changein the y value between two points bythe change in the x value between the same two points.

These two points (x1, y1) and (x2, y2) might be given, or you might find them on the graph.

If you have the equation in the form y = mx + b, the m value is the slope.

12

12

xxyy

xym

Finding the slope in our example:Here are some points we know, either from the graph or from memory:

(0°C, 32°F)(−40°C, − 40°F)(10°C, 50°F)

595090

40104050

xym

On your own… Try this with another pair of points, or use the same points in the opposite order.As long as the points you use are on this line you will ALWAYS get 9/5 as the slope!

Finding the y-intercept:


If we have the slope m and a point on the line (x, y), sub these three values into y = mx + b and solve for b.

If you have the equation in the form y = mx + b, the b value is the y-intercept.

If you have the equation in a form other than y = mx + b, either put it into y = mx + b form or sub-in x = 0 and solve for y. This answer is the y-intercept b.

Finding the y-intercept in our example:.So far we know that the slope is 9/5 and we know a point on the line (10, 50).

Now we can sub-in:

m = , x = 10, y = 50

bb

b

bmxy

321850

105950

On your own… Try this with another point. Remember though that m won’t change because it is a constant for this particular line. As long as the point you choose is on this line and use m = 9/5, you will ALWAYS get 32 as the y-intercept!

59

USING THE EQUATION OF A LINEThe temperature in Dallas, Texas is 94°F.What is that temperature in degrees Celsius?

We have found m = and b = 32,

so we have the following equation:

or

This is the equation of the line in the graph relating degrees Celsius (x or C) to degrees Fahrenheit (y or F).

So, when F = 94°F, we sub this into the equation and calculate that C = 34.4°C

59

3259

xy 3259

CF

The temperature in Dallas, Texas is 94°F. What is that temperature in degrees Celsius?

USING THE EQUATION OF A LINE

(34.4, 94)

Cell Phone Bill – A linear functionWhat are two variables involved in simple cell phone billing

system?• monthly usage (minutes) – x because this is the variable

you can directly influence (independent variable)• monthly bill amount (dollars) – y because it depends on x

(dependent variable)

What are two constants involved in simple cell phone billing system?

• flat/base monthly fee (dollars) – b because this is the initial value

• price per minute used (dollars/minute) – m because it is the rate

Let’s say that it costs 20 cents per minute and that you are always charged a monthly fee of $7.00.

Questions:1) Give the function that

relates the two variables(x – number of minutes,y – monthly bill).Draw the graph of thisrelationship

2) If you talked for 45 minutes,what will your bill be

3) If your bill is $37.40, for howmany minutes were youon the phone?

Cell Phone Bill – A linear function

2) If x = 45minutes, y = $163) If y = $37.40, x = 152minutes

Answers:1) y = 0.20x + 7

On another plan, in January you talked on your phone for 100 minutes and your bill was $30.00. In February you talked for 150 minutes, and your bill was $42.50.

Questions:1) What is the charge per

minute, and is the flatmonthly flat fee?

2) Give the function that relatesthe two variables. Draw thegraph of this relationship.

3) If you talked for 45 minuteson this plan, what will yourbill be?

Cell Phone Bill – A linear function

2) y = 0.25x + 53) If x = 45minutes, y = $16.25

Answers:1) m = 0.25$/minute b = 5.00$

But what if we want to know when these two plans cost the same amount?

We will combine the two equations into a system.A system of linear equations is a set of two simultaneous equations.

The solution to a system is the point (x, y) at which both equations hold true.

Systems of Equations

Graphically this is the intersection of the two lines.

There are infinitely many x, y pairs which satisfy the equation3x + 4 = y:(1, 7) or (0, 4) or (−1/3, 3) or (−100, −296) just to name a few…(this is the same as saying there are infinitely many points on the liney = 3x + 4)

…but if y = −7x − 1 must ALSO be satisfied, then none of the points listed work; none satisfy BOTH equations.(the only point that satisfies both equations is the point of intersection of the two lines)

So then what is the solution to ?

1743

xyxy

Systems of Equations

BIG IDEA: Turn 2 equations with 2 unknowns (hard to solve)into 1 equation with 1 unknown (easy to solve)

This brace indicates theequations form a SYSTEM

At the point of intersection, both lines will have the same y value.So we can replace the y in one equation by the equivalent value of y from the other.

y = 3x + 4(– 7x – 1)= 3x + 4 –10x = 5 x = –0.5

Half way there!!

Now with this half of the solution we can find the other variable. It doesn’t matter which original equation you choose:

y = 3(–0.5) + 4y = 2.5

OR y = –7(–0.5) -1 y = 2.5

The same!

Solving Systems by Substitution

Therefore, the solutionto

is (−0.5, 2.5)

1743

xyxy

This is the ONLY (x, y) pair that satisfies BOTH equations!

1743

xyxy

Example: Solve by substitution:

Solving Systems by Substitution

31353112

yxxy

1) Isolate one variable in one equation.(choose wisely!)

23113112

xy

xy

2) Substitute this expression into the other equation

312

31135

3135

xx

yx

3) Solve for the remaining variable

59519629331062311310

312

31135

xxxx

xx

xx

Solution:

4) Use one of the original equations to solve for the second variable 2

2)5(311

2311

xy

Let’s double check our answer. The solutionx = 5 and y = −2 should satisfy both equations:

Both are satisfied, so our solution is correct!

The solution is (5, −2)

4415114

)5(311)2(23112

xy

313131)2(3)5(53135

yx

Solving Systems by SubstitutionExample cont’: Solve by substitution:

31353112

yxxy

This is another method used to solve linear systems. It eliminates one of the variables (turns a question of 2 equations and 2 unknowns into a question with 1 variable and 1 unknown) by adding/subtracting the equations.

Ex. Solve this system of equations using elimination.Let’s “mush ‘em together” (that is, let’s add the equations

Still two variables…this didn’t help!

Solving Systems by Elimination

31353112

yxxy

Um… let’s align the equations first.

31351123

yxyx

4118 yx

Let’s try to add again.


31353112

yxxy

Let’s add the equations.

Let’s try multiplying the equations through be a number so their coefficients are opposite before addition

Let’s multiply the first by 3

626103369

yxyx

Let’s multiply the second by 2

95019 yx Let’s add them now...

Success! We eliminated y, and are left with 1 equation with 1 unknown (x), which is easy to solve!

5x

Half way there!!


To find y, simply plug-in x = 5 into either of the original equations:

31353112

yxxy

242

1511253112

3112

yyyy

xy

26331325313553135

yyyyyx

31353112

yxxy

The solution to is (5, −2)

Example: Solve by elimination:


1) Multiply and align (get two coefficients to be opposite)

2) Add to eliminate one variable

3) Solve for the remaining variable

Solution:4) Use one of

the original equations to solve for the second variable

1192242

hfhf

1313

1192242

h

hfhf 1

1313

hh

112

11212

fff

hf

The solution is h = −1, f = 1

119212

hfhf

Solution:

Let E be the price of the English textbookLet M be the price of the Math textbook

Word problem example

A certain Math textbook costs $10 more than 3 times the amount of an English book, before taxes. Together they total $140, before taxes. Calculate the price of each book.

M =10 + 3EM + E = 140

140310

EMEM

The Mathematics text costs $107.50, and the English text costs $32.50

50.107$)50.32($310

310

MM

EM

50.32$1304140310140

EEEEEM

…using substitution. …using elimination:

Answers:a) (9, 4) c) (2, −3) e) (1, −3)

b) (−4, 7) d) (0.5, −0.5) f) (250, 700)

2. Both plans cost the same, $15.00 when you use 40min

2. Back to the cell phone example, how many minutes do you have to use for both cell phone plans to cost the same?

1954253

b)yx

yx

142295

d)yxyx

29

23

1542e) xy

yx

1332

953c)

yxyx

459912

a)yxyx

50.11112.011.0950

f)yx

yx

…using your choice:1.Solve these systems…

university prep math

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