mid-term review notes

MID-TERM REVIEW NOTES

DO NOT LOSE THESE!! WE WILL ADD TO THESE DAILY

REAL NUMBERS

• Real numbers are every number that can be found on a number line.

NOT A REAL NUMBER (FAKE)

• Any expression that has zero as the denominator.

REAL NUMBERS INCLUDE:• RATIONAL NUMBERS- any number that can be written as a

fraction {integers and fractions}• INTEGERS- whole numbers (counting numbers including 0)

AND their opposites (negatives) {…-3,-2,-1,0,1,2,3…}• WHOLE NUMBERS- counting numbers including zerO,

{0,1,2,3…}• NATURAL NUMBERS- counting numbers, {1,2,3…}

• IRRATIONAL NUMBERS- any number that cannot be written as a fraction {square root of a non-perfect square and pi}

• Ray – rational numbers ARE…• Found – fractions• Me – mixed numbers• Packing – percents %• Ice – integers• In – improper fractions• The – terminating decimals• Restaurant – repeating decimals

Ray Found Me Packing Ice In The Restaurant

REAL NUMBERS

SYSTEMS OF EQUATIONS

• A system of equations is when you have two or more equations using the same variables.

• The solution to the system is the point that satisfies ALL of the equations. This point will be an ordered pair, no solution, or infinitely many solutions.

• Three different methods: Graphing, Substitution, Elimination

Review: Graphing with slope-intercept

1. Start by graphing the y-intercept (b = 2).

2. From the y-intercept, apply “rise over run” using your slope.rise = 1, run = -3

3. Repeat this again from your new point.

4. Draw a line through your points.

• M = - 1/3• B = 2

1-3

Start here

1-3

Y = - 1 X + 2 3 GRAPHING

Intersecting Lines

• The point where the lines intersect is your solution.

• The solution of this graph is (1, 2)

(1,2)

GRAPHING

Parallel Lines

• These lines never intersect!

• Since the lines never cross, there is NO SOLUTION!

• Parallel lines have the same slope with different y-intercepts.

2Slope = = 21

y-intercept = 2y-intercept = -1

GRAPHING

Coinciding Lines

• These lines are the same!

• Since the lines are on top of each other, there are INFINITELY MANY SOLUTIONS!

• Coinciding lines have the same slope and y-intercepts.

2Slope = = 21

y-intercept = -1GRAPHING

SYSTEMS OF EQUATIONS POSSIBLE SOLUTIONS:

If you solve using substitution or elimination

X and Y can be an ordered pair. X = 4, Y= 7. Answer: (4,7) ONE SOLUTION

If you solve, and the variables cancel out, leaving you 8 = 8; This is a true statement therefore, Answer: INFINITELY MANY SOLUTIONS.

If you solve, and the variables cancel out, leaving you 8 = 0; This is NOT a true statement therefore, Answer: NO SOLUTION.

Solving a system of equations by substitution

Step 1: Solve an equation for one variable.

Step 2: Substitute

Step 3: Solve the equation.

Step 4: Plug back in to find the other variable.

Step 5: Check your solution.

Pick the easier equation. The goalis to get y= ; x= ; a= ; etc.

Put the equation solved in Step 1into the other equation.

Get the variable by itself.

Substitute the value of the variableinto the equation.

Substitute your ordered pair intoBOTH equations.

SUBSTITUTION

SOLVE USING SUBSTITUTION 3x – y = 4 x = 4y – 17

SUBSTITUTION

Solving a system of equations by elimination using addition and subtraction.

Step 1: Put the equations in Standard Form.

Step 2: Determine which variable to eliminate.

Step 3: Add or subtract the equations.

Step 4: Plug back in to find the other variable.

Step 5: Check your solution.

Standard Form: Ax + By = C

Look for variables that have thesame coefficient.

Solve for the variable.

Substitute the value of the variableinto the equation.

Substitute your ordered pair intoBOTH equations.

ELIMINATION

SOLVE USING ELIMINATION x + y = 53x – y = 7

ELIMINATION

Solving a system of equations by elimination using multiplication.

Step 1: Put the equations in Standard Form.

Step 2: Determine which variable to eliminate.

Step 3: Multiply the equations and solve with addition or subtraction.

Step 4: Plug back in to find the other variable.

Step 5: Check your solution.

Standard Form: Ax + By = C

Look for variables that have thesame coefficient.

Multiply both equations and solve for the variable.

Substitute the value of the variableinto the equation.

Substitute your ordered pair intoBOTH equations.

ELIMINATION

SOLVE USING ELIMINATION2x + 2y = 63x – y = 5

ELIMINATION

SOLVE F(X) = G(X)

• F(X) = 2X + 4

• G(X) = 6X – 8

• Set the equations equal to each other and solve for x.

F(X) = G(X)2X + 4 = 6X – 8-2X -2X 4 = 4X – 8 +8 +8 12 = 4X 4 4 3 = X

F(X) = G(X)

Slope (M)

• A measure of the steepness of a straight line• Tells how fast one variable changes compared

with the other.• Rise over run

1212xxyyM

BAM

3) Find the slope of the line that goes through the points (-5, 3) and (2, 1).

m y2 y1

x2 x1

1 32 ( 5)

m

1 32 5

m

27

m

SLOPE

Determine the slope of the line.

The line is decreasing or going down the hill (slope is negative).

2

-1

riserun

2 1

2

Find points on the graph. Use two of them and apply rise over run.

SLOPE

When an equation is in slope-intercept form:

What is the slope? ____________

2 1y x Now look at the equation below……

What is the intercept? ____________ SLOPE

Find the x- and y-intercepts of x - 2y = 12

x-intercept: Plug in 0 for y.x - 2(0) = 12x = 12; (12, 0)

y-intercept: Plug in 0 for x.0 - 2y = 12y = -6; (0, -6) SLOPE

You can also find slope when given a table of

values.X Y

1 4

2 5

3 6

Pick any two points and find the slope.

(1,4) and (2,5)

m = y2 – y1

x2 – x1

M = ( 5 – 4 ) ( 2 – 1 )

M = 1 = 11 SLOPE

Types of Slope

Positive Negative

ZeroUndefined

or No Slope

SLOPE

Remember the word “VUXHOY”V=vertical linesU=undefined slopeX=number; This is the equation of the line.

H=horizontal linesO=zero is the slopeY=number; This is the equation of the line.

SLOPE

y

Tell whether the graph is linear or nonlinear.

A. B.

The graph is a straightline, so the graph is linear.

The graph is not a straight line, so it is nonlinear.

x–4

–4

4

4

0x

–4

–4

4

4

0

y

Course 2

12-6 Nonlinear Functions

Tell whether the function in the table has a linear or nonlinear relationship.A.

difference = 3difference = 6

difference = 1difference = 1

The difference between consecutive output values is not constant.

The difference between consecutive input values is constant.

The function represented in the table is nonlinear.

Input Output1 22 53 11

Course 2

Nonlinear Functions

Tell whether the function in the table has a linear or nonlinear relationship.

Example 2B: Identifying Nonlinear Relationships in Function Tables

A.

difference = 3difference = 3

difference = 1difference = 1

The difference between consecutive output values is constant.

The difference between consecutive input values is constant.

The function represented in the table is linear.

Input Output1 32 63 9

Course 2

12-6 Nonlinear Functions

Components of a Graph• Title

– Every graph must have a title

• Subtitles– Explains what the horizontal and vertical quantities

represent

• Equally spaced divisions

Graphs are used to present numerical information in picture form.

• Scatter Plot is a graph of two related sets of data on an XY axis.

• These are useful when you want to study related pairs, such as height and weight.

• Correlation is the relationship between two or more things.

• Linear Correlation is a scatter plot that forms a “line” showing that one axis seems to depend on or relate to the other.

Line of Best Fit• Line of best fit- line that seems to describe the direction

the points are heading in.

• There are methods for determining where this line is, there are two criteria to finding and drawing the line:– The line of best fit must more or less follow the direction of

the points.– There should be roughly the same number of points on each

side of the line.

• Lines of best fit can be used to predict results, especially if you find the line's equation.