
INTERMEDIATE ALGEBRANotes Semester 1IAUnit5 PolynomialsDay 1 Number Sets and PropertiesIntAlg1Notes2 Solving EquationsIntAlg1Notes3 Absolute Value EquationsA scene in the movie Smilla's Sense of Snow compares human life to the number system:Properties of EqualityNotes 2Solving EquationsWhat are the order of operations PEMDAS?There are two kinds of equationsarithmetic and algebraic.The first is easy to solve. Just do what it says: Add 5 to 3:The second is more difficult. What number plus 2 is 6?You can guess, or use the Properties of Equality to isolate the variable, m.Do the opposite of adding 2. Subtract 2. m = 6  2 so m is 4.This is called a onestep equation. Here's another:Do the opposite of subtracting 3 to isolate x.Add 3 to both sides using Addition Property.Use the Additive Identity to get rid of 0.Here's a one step equation using the Division Property.This example is a twostep equation.Here's another equation that requires two steps to solve.Since the equation says multiply x by 4 first, then add 8 to get 24.Do the inverse operations in reverse order. Subtract 8, then divide by 4.What happens to n? Multiplied by 5 then subtract 11.Inverses in opposite order to 24: Add 11 then divide by 5.Sometimes equations take many steps to solve. These are multistep equations.How did they ditch the 6? Subract 6.How did they ditch the 8? Divide by 8.What if the distributive property is involved?Now it's a multistep equation. Isolate x on the right by subtracting 5x from both sides then adding 5.Try this example:Here's one with a fraction:Now there's a fraction. It would have been better to multiply by 4 first:3(x+3) = 9*43x + 9 = 363x = 369x = 27/3x = 9Here's a multistep distributive equation with a fraction.
This one has a double distributive property:Day 3Absolute Value EquationsThese techniques are useful in absolute value equations:Note there are two solutions. By definition of absolute value either x+2 = 7 or its opposite does: (x+2)=7.First, solve x = 7  2. There are two ways to go about the second part:x  2 = 7 OR x + 2 = 7.Both give the same answer:x = 9 OR x = 9.How can you be sure to find both solutions?The inside could be 7 or 7. So we have two equations:x  2 = 7 OR x  2 = 7x = 7+2 OR x = 7+2x = 9 or x = 5.Review of absolute value definition:and as a fuction.In other words, if x is negative, make it positive.If x is positive or zero, keep it the same.Unit 2 Chapter 3Graphing Linear EquationsIntAlg2Notes1 Graphing Linear EquationsIntAlg2Notes2 Function NotationIntAlg2Notes3 SlopesIntAlg2Notes4 Slope InterceptsIntAlg2Notes5 Functions, Relations, Vertical Line TestIntAlg2Notes6 Vertical, Horizontal, Parallel, Perpendicular LinesIntAlg2Notes7 Linear Inequalities
Graphing Linear EquationsA Cartesian grid (named after Rene DesCartes) is formed when two number lines intersect to form a right angle.The intersection of the lines is called the origin.The horizontal number line is the xaxis and the vertical line is the yaxis.Moving right on the xaxis corresponds to a positive number. A negative number is to the left of the origin.On the yaxis, positive numbers mean moving up from the origin, and negative numbers mean moving down.So (2, 5) means from the origin move right 2 units and up 5. This location point lies in Quadrant I.Moving left 3 and up 6 would be (3, 6) and places the point in the second quadrant, QII.(1, 1) is in the third quadrant (QIII), and (2, 3) would be in the fourth (QIV).To graph a line, there are several forms of equations.For the slopeintercept form, y = mx + b, ordered pairs can be generated in a table.To graph a line, say y = 3x+1, only two points are needed.So evaluate for x = 0 and x = 1.3(0) +1 = 1 and 3(1) + 1 = 4. Graph (0,1) and (3, 4) then draw the line.To graph an equation in standard form, Ax + By = C, plug 0 into each variable and solve for the other.Let x = 0 and solve 3(0) + 2y = 6 for y. Then substitute y = 0 and solve 3x + 2(0) = 6 for x.By graphing the points (2,0) and (0,3) we can graph the line and also locate both intercepts.Graph x  2y = 2.Here are steps for graphing the slope intercept form.Here is another form called pointslope.Here's a formula map for all linear equations.Function NotationDEFINITION: A "function" is a rule or correspondence that maps one number (or input element) to exactly one output value. If there are two possible outputs for the same input, the correspondence is not a function and is called a "relation" instead.Find the rule that maps the Domain numbers into the Range bubble:The rule is to square the input number. f(4) = 16 because 4*4 is 16. f(x) = x^2.We say that as "f of x equals x squared."You can think of a function as a box or a room where something happens to the number going in.In this case, 1 is added to the input number, and the output is the sum.In general,The input is called the Domain and the output is the Range.What are the domain and range values in the mapping below.Domain: {4, 5, 8} Range: {2, 5, 7, 9}Note this example is not a function as the input 5 has two different output values, 2 and 9.Still, each relation can be written as an ordered pair: (4, 7); (5, 2); (5, 9); (8, 5)If the relation is a function, the rule can be written as an equation.Function notation can also be written as an equation starting with "y =".Now the equation can be described as a rule of correspondence.For example, this rule is to multiply by 3 and then add 5.
SLOPES and LINEAR EQUATIONSNotes reviewing linear equations (need not copy systems examples.)
There's a typo in #5. It should read "Up 1, Left 4" as it is a negative slope.Graphing Slopes/Slope InterceptsFunctions and RelationsWrite your name in the first input bubble. Then count the number of vowels and draw an arrow to that number in the second bubble.This is a mapping diagram for a function. Each name has exactly one number of vowels in it.As each input value corresponds to one output value, this relation is a function.However, each output value can come from more than one input.For example, two different names can have the same number of vowels.But no name can have two different number of vowels in it.Is every relation mapping one set of values to another a function? Not necessarily.Function Rule: Map your birthday month to the sum of digits in your birthday.For example, August 19 would map August > 10 because 1+9 = 10.Convert the month to their number (Jan = 1, Feb = 2, ... ) and convert the birth data into ordered pairs.For example, March 14 becomes (3, 5) because March is the 3rd month and 1+4 = 5.Now graph the ordered pairs. Note that the repeated months prevent this rule from being a function.For example, there are two birthdays in December so the input x = 12 has two outputs, y = 4 and y = 6.Graphing repeated xvalues with different youtput values has points vertically above each other.This is an example of the relation failing the vertical line test. So this rule is NOT a function.The set of values for the input is called the Domain. The set of output values is called the Range.What are the Domain and Range for this data set?DOMAIN: {Feb, Mar, May, July, Sept, Oct, Nov Dec}RANGE: {1, 2, 3, 4, 5, 6, 7, 9, 11}Mathematically, what could the smallest number for the range be? The largest number?Convert the data above into ordered pairs (month, digit sum).The circled pairs indicate repeated input values mapping to different output values.So this is not a function.The repeated input values when graphed fail the vertical line test.SPECIAL LINEAR EQUATIONSVertical and Horizontal Lines, Parallel and Perpendicular Lines
IntAlg2Notes 7Graphing InequalitiesVertical lines are of the form x = k.What do the graphs of x < k or x > k look like?The solution of ordered pairs (x,y) that satisfy the inequality is called a halfplane.Which graph is the true graph of y > 1 ?Graph 2 is y > 1. The others are y < 1, y < 1, and y > 1.For linear equations in slopeintercept form form y = mx + b, which side do we shade?If the inequality is > or >, shade the upper halfplane.(y > mx+b, y > mx+b)Shade the lower halfplane if the inequality readsy < mx+b or y < mx + b.Here are the steps to graph y < (3/2)x + 3.Use a solid line for "or equals to" symbols.Shade below, or check the point (0, 0). Is 0 < (2/3)0 + 3?Yes, 0 < 3. Shade on side of origin.Graph y > (1/2)x 3. Use a dashed line when graphing y = (1/2)x  3 since it is a strict inequality.Graph y > 2x 5.Graph 2x + 5y > 5. First put inequality in slope intercept form.Graphing standard form inequality Ax + By < CTest points for graphing x + y < 3.So shade below on the same side as point (1, 2).Why can't (0, 0) be used as a good testing point for graphing this inequality?
Unit 3 Chapter 4Systems of EquationsIntAlg4Notes1 Solve by GraphingIntAlg3Notes2 Solve by SubstitutionIntAlg3Notes3 Solve by EliminationIntAlg2Notes4 Solve with TechnologyIntAlg4Notes5 Solving with MatricesSolving with Cramer's RuleLesson 1  Solving Systems by Graphing
SOLVE Systems by Substitution(p226 evens 1622 done as class notes)System EliminationUse this online software to check answers and see solutions:Solving Systems with TechnologyGRAPHING CALCULATORS:To graph the system of equations, follow these steps:Press Y= .Enter first equation into Y1. Press ENTER.Enter 2nd equation into Y2.Press ZOOM and then press 6 : ZStandard.Or arrow to 6 and press ENTER.(This sets the xyaxes on 10 to 10.)A graphing window should appear.To find the point of intersection:Press 2nd TRACE to access the CALC menu.Select or press 6 : Intersect.The calculator asks "First curve?" and if the cursor is on Y1 press ENTER.Repeat to answer "Second curve?" and ENTER.To answer "Guess?" arrow the cursor to the point of intersection and press ENTER.The bottom left of the screen says "X = " and the bottom right "Y = " with the solution.ALSO, on the internet check out this website online:It lets you enter equations in any form (Ax+By=C, y=mx+b, yy1=m(xx1), etc.).Separate each equation by a semicolon ";" and it SOLVES and GRAPHS it for you.
SYSTEM WORD PROBLEMSHow many pennies are there in circulation?
Matrix Row Operations
Solve these with Matrix Row Operations:Let's do the two with fractions with graphing calculators.Write system #27 as a matrix:Now enter it into a TI graphing calculator.Press 2nd MATRIX, and arrow to "EDIT". Press ENTER or 1 to edit Matrix A.Press 2 ENTER and 3 ENTER to set the size of matrix A to 2 rows and 3 columns (2x3).Enter each fraction and press ENTER between each to fill in the matrix.Then press 2nd QUIT, 2nd MATRIX, ENTER, MATH 1: Frac ENTER.[A] Math→FracNow to solve it. Press 2nd MATRIX, move to "MATH", move down to A:ref(, and press ENTER.Then press 2nd MATRIX, ENTER or 1, press MATH and ENTER or 1:>Frac to display in fractions.Ans → FracThe is called the "reduced echelon form" and it tells us that 0x + 1y = 72, or y = 72.To find x we can back substitute, or repeat the process above this time with B:rref( from the MATRIX MATH menu.Press 2nd MATRIX, move to "MATH", move down to B:rref(, and press ENTER.Then press 2nd MATRIX, ENTER or 1, and ENTER again.orThis form is called "reduced row echelon form" and tells us that 1x + 0y = 34, or x = 34, and that to y = 2.Now let's do #28 with the rref command. First, enter the system into matrix A.Now Press 2nd MATRIX, move to EDIT, select B: rref(, 2nd MATRIX ENTER.rref([A]It displayed it in decimals so let's frac it!MATH 1: Frac ENTERrref(Ansx = 1/7 and y = 75/14 or 5 5/14.
3by3 SystemsContinue until you get a 3by3 diagonal of 1s with the other entries 0s:If you end up with an entire row of 0s, then the system has infinitely many solutions and is dependent.Cramer's Rule3 by 3 systems
3 by 3 DeterminantsCopy the first two columns to the right:Now draw diagonals through the numbers:Add the downward diagonal products, and subtract the upward diagonal products.Evaluate the products, sum, and difference:If a determinant is 0, then the system is either inconsistent or dependent,indicating its matrix is not invertible.POLYNOMIALS
Last Modified on January 6, 2016