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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
variablea 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
23x
xx
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
612
1g
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 Functions
The 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….
325
9 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.
THE EQUATION OF A LINE
m = 1
m = 3
m = −1/2
m = 0
THE EQUATION OF A LINE
The 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).
THE EQUATION OF A LINE
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.
THE EQUATION OF A LINE
Finding slope:
THE EQUATION OF A LINE
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
xx
yyx
ym
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)
5
950
90
4010
4050
x
ym
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:
THE EQUATION OF A LINE
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
b
b
b
bmxy
32
1850
105
950
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!
5
9
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
5
9
325
9 xy 32
5
9 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 ?
17
43
xy
xy
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)
17
43
xy
xy
This is the ONLY (x, y) pair that satisfies BOTH equations!
17
43
xy
xy
Example: Solve by substitution:
Solving Systems by Substitution
3135
3112
yx
xy
1) Isolate one variable in one equation.(choose wisely!)
2
311
3112
xy
xy
2) Substitute this expression into the other equation
312
31135
3135
xx
yx
3) Solve for the remaining variable
5
9519
6293310
62311310
312
31135
x
x
xx
xx
xx
Solution:
4) Use one of the original equations to solve for the second variable 2
2
)5(311
2
311
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)
44
15114
)5(311)2(2
3112
xy
3131
31)2(3)5(5
3135
yx
Solving Systems by Substitution
Example cont’: Solve by substitution:
3135
3112
yx
xy
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
3135
3112
yx
xy
Um… let’s align the equations first.
3135
1123
yx
yx
4118 yx
Let’s try to add again.
Solving Systems by Elimination
3135
3112
yx
xyLet’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
62610
3369
yx
yx
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!!
Solving Systems by Elimination
To find y, simply plug-in x = 5 into either of the original equations:
3135
3112
yx
xy
2
42
15112
53112
3112
y
y
y
y
xy
2
63
31325
31355
3135
y
y
y
y
yx
3135
3112
yx
xyThe solution to is (5, −2)
Example: Solve by elimination:
Solving Systems 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
1192
242
hf
hf
1313
1192
242
h
hf
hf 1
1313
h
h
1
12
112
12
f
f
f
hf
The solution is h = −1, f = 1
1192
12
hf
hf
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
140
310
EM
EM
The Mathematics text costs $107.50, and the English text costs $32.50
50.107$
)50.32($310
310
M
M
EM
50.32$
1304
140310
140
E
E
EE
EM
…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?
1954
253b)
yx
yx
142
295d)
yx
yx
2
9
2
31542
e)xy
yx
1332
953c)
yx
yx
4599
12a)
yx
yx
50.11112.011.0
950f)
yx
yx
…using your choice:1.Solve these systems…