University Prep Math Patricia van Donkelaar Course Outcomes By the end of the course, students should be able to 1)Solve linear equations 2)Solve a system of equations 3)Identify linear, quadratic and exponential patterns 4)Algebraically find the equation of each type of pattern 5)Graph linear and quadratic functions 6)Transfer between the 3 forms of a quadratic 7)Solve quadratic equations by factoring, completing the square or by using the quadratic root formula 8)Solve for the vertex of a quadratic 9)Solve exponential equations using common bases and logs 10)Solve logarithmic equations 11)Solve simple probability problems 12)Use permutations and combinations to solve problems involving probability

DefinitionExample 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. expression a 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. equationtwo 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 or finding 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 equation 2x – 6 = 4 would give a solution (root) of x = 10. Solving the equation x 2 = 4 gives roots 2 and –2

Order of Operations: When to do what RACKETSRACKETS XPONENTSXPONENTS IVISIONIVISION ULTIPLACTIONULTIPLACTIOND DDITIONDDITION UBTRACTIONUBTRACTION B E D M A S

Evaluate the following expressions given the value of the variable stated. 1) 7x – 3if x = 7 2) 10(x – 2)if x = 4 3) 5r – 7t –6if r = 2 and t = 1 4) 3t 2 +5t – 9if t = 2 5) if x = 4 6) if j =3 Answers: 1) 46 2) 20 3) – 3 4) 13 5) 26 6) 1

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) x = 26 Answers: 1) 6 2) –5 3) 11 4) 10/6 (or 1.666…) 5) 26 6) –3 7) 5 and –5

Functions A 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 = x 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 relationship between f (liters of fuel) and p (price) The simplest functions are linear linernot 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….

The equation y = mx + b describes 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 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 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 = 3b =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 these two 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 much change there was for y (the dependent variable) for every change in x (the independent variable). Mathematically we divide the change in the y value between two points by the change in the x value between the same two points. These two points (x 1, y 1 ) and (x 2, y 2 ) 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.

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) 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 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!

The 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

The temperature in Dallas, Texas is 94°F. What is that temperature in degrees Celsius? (34.4, 94)

Cell Phone Bill – A linear function What 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 this relationship 2)If you talked for 45 minutes, what will your bill be 3)If your bill is $37.40, for how many minutes were you on the phone? Cell Phone Bill – A linear function 2) If x = 45minutes, y = $16 3) 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 $ In February you talked for 150 minutes, and your bill was $ Questions: 1)What is the charge per minute, and is the flat monthly flat fee? 2)Give the function that relates the two variables. Draw the graph of this relationship. 3)If you talked for 45 minutes on this plan, what will your bill be? Cell Phone Bill – A linear function 2) y = 0.25x + 5 3) 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 equation 3x + 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 line y = 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 ? 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 the equations 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) + 4 y = 2.5 OR y = –7(–0.5) -1 y = 2.5 The same! Solving Systems by Substitution Therefore, the solution to is (−0.5, 2.5) This is the ONLY (x, y) pair that satisfies BOTH equations!

Example: Solve by substitution: Solving Systems by Substitution 1) Isolate one variable in one equation. (choose wisely!) 2) Substitute this expression into the other equation 3) Solve for the remaining variable Solution: 4) Use one of the original equations to solve for the second variable

Let’s double check our answer. The solution x = 5 and y = −2 should satisfy both equations: Both are satisfied, so our solution is correct! The solution is (5, −2) Solving Systems by Substitution Example cont’: Solve by substitution:

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 Um… let’s align the equations first. Let’s try to add again.

Solving Systems by Elimination 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 Let’s multiply the second by 2 Let’s add them now... Success! We eliminated y, and are left with 1 equation with 1 unknown (x), which is easy to solve! Half way there!!

Solving Systems by Elimination To find y, simply plug-in x = 5 into either of the original equations: The 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 The solution is h = −1, f = 1

Solution: Let E be the price of the English textbook Let 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 + 3E M + E = 140 The Mathematics text costs $107.50, and the English text costs $32.50

…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? …using your choice: 1.Solve these systems…