Steve Greer sgreer@hrsb.ca hrsbstaff.ednet.ns.ca/sgreer University Prep Math Steve Greer sgreer@hrsb.ca hrsbstaff.ednet.ns.ca/sgreer Course Outcomes By the end of the course, students should be able to Solve linear equations Solve a system of equations Identify linear, quadratic and exponential patterns Algebraically find the equation of each type of pattern Graph linear and quadratic functions Transfer between the 3 forms of a quadratic Solve quadratic equations by factoring, completing the square or by using the quadratic root formula Solve for the vertex of a quadratic Solve exponential equations using common bases and logs Solve logarithmic equations Solve simple probability problems Use permutations and combinations to solve problems involving probability

University Prep Math Phrase or Term Definition Example A variable The letter which stands for a number that can change. ‘x’ in 2x-6 is a variable. Its value could be 4, or -6, or any other number. A function The relationship between two variables. y = 2x is a relationship between two variables (x and y). In this case, the function says “y is always twice as much as x”. Evaluating an expression Finding out what the value of the expression is when you know the value of the variable. Evaluate: 4x-1 when x = 3. We plug in a 3 wherever we see an ‘x’ and follow the order of operations. Phrase or Term Definition Example Phrase or Term Definition Example A variable The letter which stands for a number that can change. ‘x’ in 2x-6 is a variable. Its value could be 4, or -6, or any other number. Phrase or Term Definition Example A variable The letter which stands for a number that can change. ‘x’ in 2x-6 is a variable. Its value could be 4, or -6, or any other number. A function The relationship between two variables. y = 2x is a relationship between two variables (x and y). In this case, the function says “y is always twice as much as x”.

Order of Operations: When to do what B E D M A S I V S ON DD I T ON R A C K E T S X P ON E N T S U B T R A C I ON U L T I P A C ON

“Evaluating an Expression” examples Evaluate the following expressions given the value of the variable stated. 7x-3 if x = 7 2. 10(x-2) if x = 4 5r -7t -6 if r = 2 and t = 1 4. 3t2 +5t -9 if t = 2 3x(x-2) + x/2 if x = 4 6. if j =3 Answers: 1. 46 2. 20 3. -3 4. 13 5. 26 6. 1

Find the root of each equation. 1) 5(x-4) = 10 2) 8w – 2 = -42 Phrase or Term Definition Example Solving an equation or finding the roots. Finding out what the value of the variable is when you know what the expression equals. This is the opposite as evaluating the expression. Root – the value of the variable that makes the equation true. Solve for ‘x’. 3x-4 = 8 We will “undo” BEDMAS Answers: 6 2) -5 3) 11 4) 10/6 = 1.67 5) 26 6) -3 Find the root of each equation. 1) 5(x-4) = 10 2) 8w – 2 = -42 3) ½ (g+1) = 6 4) 3x + 6 = 9x – 4 5) 6)

Where do these functions come from? We have worked with a few expressions and a few equations but what do they mean? Let’s look out how our every day lives use functions all the time!

Linear Function: Temperature The temperature in Dallas, Texas is 940F. What is that temperature in degrees Celsius?

Here’s a graph showing the relationship between the two temperature measurements. If we can describe this graph mathematically, we can calculate the temperature in Dallas. We need the EQUATION OF A LINE

y = mx+b describes ANY (straight) line. THE EQUATION OF A LINE This equation has two parts. The slope, which is given the symbol ‘m’ (DUH!) and the y-intercept, given the symbol (you guessed it) ‘b’ y = mx+b describes ANY (straight) line.

THE EQUATION OF A LINE PART 1: The slope of a line is a number value indicating how steep the line is. Each of these lines is as steep as the others. They have the same slopes.

THE EQUATION OF A LINE PART 2: The vertical (y) intercept is the point where the line crosses the vertical (y) axis. These lines share the same vertical intercept. But they each have a different slope.

THE EQUATION OF A LINE Each distinct line has a specific slope and a specific y-intercept. If these are defined, then we can find every point on the line. Finding slope: The slope indicates how much change there was in the dependant variable (y) for every change in the independent variable (x). Mathematically, that means to find the CHANGE in the ‘y’ variable and divide it by the CHANGE in the ‘x’ variable. Slope: Δy / Δx We can use two points on the graph to calculate the slope, or we can identify it as a RATE OF CHANGE given in the question.

Finding the slope.

THE EQUATION OF A LINE Each distinct line has a specific slope and a specific y-intercept. If these are defined, then we can find every point on the line. Finding the y-intercept EVERY y-intercept can be given in the form (0,y). This means that if we let x = 0, we can get the y-value and thus the y-intercept. Mathematically, that means to find the VALUE of the ‘y’ variable when the ‘x’ variable has a value of 0. Let’s find the y-intercept of our equation.

USING THE EQUATION OF A LINE Our equation is F = 9/5C + 32 We know the value of “F”, the number of degrees Fahrenheit. We want to know “C”, the number of degrees Celsius. So we need to solve the equation for C.

(34.4,94)

Cell Phone Bill – A function What are two variables involved in cell phone use? We can find a relationship between the cost of the bill and the time of usage. Which variable depends on the other? Let’s say that it costs 20 cents per minute and that you are always charged a monthly fee of $7.00. Questions: 1) What is the function? 2) If you talked for 45 minutes, what will your bill be? 3) If your bill is $37.40, how many minutes were you on the phone?

Substitution and Elimination We will now look at the situation where we compare two linear equations. This can help us figure out when two cell plans cost the same amount, for instance. We can use elimination or substitution to solve a system of equations. A set of equations that have common variables. In solving a system of equations, we try to find values for each of the unknowns that will satisfy every equation in the system. For example: If 3x + 4 = y, there are an infinite number of solutions to this equation. BUT if 3x + 4 =y AND -7x-1 =y, then none of those solutions work. That is, they don’t satisfy BOTH equations. (1,7) or (0,4) or (-1/3,3) or (-100,-296) to name a few

So what is the solution to 3x + 4 =y and -7x-1 =y? If both 3x + 4 and -7x-1 are equal to ‘y’, they must be equal to each other. 3x+4 = -7x – 1 5 = -10x x = -0.5 We can now use this value in either of the original equations to solve for y: y= 3 (-0.5) +4 y= 2.5 OR y = -7(-0.5) -1 y = 2.5 This is the ONLY x value that will give us the same y value in each equation The same!

Substitution Practice Ex. What values of x and y satisfy both of these equations? 3x + 2y = 11 and 5x – 3y = 31 Solve one equation for one of the variables. 3x + 2y = 11 2y = 11-3x y = (11-3x)/2 5x – 3[(11-3x)/2] = 31 Now solve for x. Substitute this expression into the other equation. 5x – 33/2 +9x/2 =31 10x/2 +9x/2 = 31 + 33/2 19x/2 = 62/2 + 33/2 19x/2 = 95/2 19x = 95 X = 5 y = (11-3(5))/2 Y = -4/2 = -2

Substitution Practice Ex. What values of x and y satisfy both of these equations? 3x + 2y = 11 and 5x – 3y = 31 Let’s double check our answer. The solution x = 5 and y = -2 will work in both equations 3(5) + 2(-2) = 11 15+ -4 = 11 11= 11 5(5)-3(-2) =31 25 + 6 = 31 31 = 31 Both are true statements!

Elimination Practice To eliminate one of the variables, we may need to multiply an equation by a factor before we add or subtract the equations. Ex. Solve this system of equations using elimination. 6h + 3f = -3 9h -2f = -11 Let’s subtract the equations. -3h + 5f = 8 Still two variables…this didn’t help!

Elimination Practice To eliminate one of the variables, we may need to multiply an equation by a factor before we add or subtract the equations. Ex. Solve this system of equations using elimination. 6h + 3f = -3 9h -2f = -11 Let’s add the equations + Let’s subtract the equations. 15h+ f = -14 Still two variables…this didn’t help!

Elimination Practice To eliminate one of the variables, we may need to multiply an equation by a factor before we add or subtract the equations. Ex. Solve this system of equations using elimination. 6h + 3f = -3 9h -2f = -11 2( 3( ) Let’s subtract the equations. Let’s add the equations Now they become 12h + 6f = -6 27h – 6f = -33 Let’s multiply the first equation by 2 and the second equation by 3 so that both have the term ‘6f’. + Now we can add them to eliminate ‘f’ 39h = -39 h = -1 To find ‘f’ simply plug -1 for h into either equation. 6(-1) + 3f = -3 -6 + 3f = -3 3f = 3 f = 1

Word problem example A certain Math textbook costs $10 more than 3 times the amount of an English book. Together, they total $140 before taxes. Calculate the price of each book. E = price of English book M= price of Math book 10+3E + E = 140 10+4E = 140 M =10+3E M + E = 140 4E = 130 E = 130/4 E = 32.5 M =107.5

Solve these systems using substitution: x- 3y = -25 2) 5x + 9y = -2 3) 2x -4y = 15 4x + 5y = 19 2x +4y = -1 Solve these systems using elimination: 4) 3x +5y = -9 5) x = 2y + 1 6) x + y = 950 2x - 3y = 13 9x – 9y = 45 0.11x + 0.12 y = 111.50 Answers: (-4,7) 2) (0.5, -0.5) 3) (1,-3) 4)(2,-3) 5) (9, 4) 6) (250, 700)