Intermediate AlgebraClark/Anfinson

Chapter 1 Linear Functions

Ch 1 - section 1Solving/Evaluating - linear equations

Definition Linear: an equation that contains only

multiplication(division) and/or addition(subtraction) Solving: determine the values of a variable that

will make a statement true.

statement: in English – a sentence about a number in Algebra - an equation or inequality

Fundamental concepts of solving An equation or an inequality contain TWO numbers (left

side and right side) The numbers in the equation can be simplified (this

employs properties and does not change the value of the numbers)

The numbers in the equation can be CHANGED with the condition that BOTH numbers are changed the same way.

You have “solved” the equation when the sentence has been changed so that the desired variable is isolated.

You isolate the variable by REMOVING (canceling) terms and factors that are attached to it

Factors vs terms A term is any number that is added to another

number - a term can be canceled by using its opposite

A factor is any number that is multiplied by another number – a term can be canceled by using its reciprocal

Examples – changing equationsChange is a matter of choice-You can change ANY way you want as long as it is BOTH numbers (sides)

a. 5x = 10

b. 4 = -3 + y

c.

d. M + 2/5 = 9/7

e. x + 5y = 12

f. 4(x – 7) = 20

g.

Order of solving a. 2x – 9 = -3

b. -25 = 5x + 7

c.

d.

e. 7(3x – 2) = 14

f. 4q – 12 + 3q = -47

g. 2 (x – 9) + 3x = 2

TERMS ON BOTH SIDES You have not solved for x if the x appears in both

numbers.

3(x – 5 ) = 7x + 2

5(x + 8) – 2(4 – 2x) = 7 + 3(2x - 8)

Dealing with fractions- clearing the denominator

Although essentially the same process there are 2 ways to write the work for this - you need to pick one way and stick with it until you understand it – then if you want to you can try the other way to explore the mathematics

Method 1 - choose to multiply both sides by the common

denominator of ALL fraction

Method 2 – Use the identity property to change all fractions

to a common denominator and then ignore the denominator solve the numerator

  Method 1 Method 2

Applying what you know While numbers are intriguing in their own right, they would never be

studied if they were not useful in answering questions in our daily lives.

A number becomes “useful” when you attach a label to it. A variable is a number and becomes “useful” when you attach a

label to it when doing application problems you will a. Define the labels for each variable b. write a useful equation using those variables or use the given equation c. Replace variables with appropriate values d. Solve or evaluate the resulting equation e. LABEL your answer appropriately.

Example: Mark has a cell phone plan that uses the formula b = 29 + .3c to figure his bill(b) based on number of

calls(c) he makes i. Find the bill if he makes 32 calls ii. Find the bill if he makes 251 calls iii. How many calls did he make if the bill was $73

A. Identify labels for variables B. Equation given C. Replace the appropriate variable D. solve or evaluate E. Label the answer

Example 2: Mary works for a company doing sales. She

earns a monthly salary based on how much she sells. She will receive a retainer fee of $500 and 4% of her total sales.

i. Write an equation to model this situation. ii. Find her pay for July if she sells $53,562.85. iii. How much will she need to sell in November if she needs $ 3000?

literal equations; The directions “solve for x” will always mean to

find the number which can replace x and make the equation true – however this “number” is not always a numeral – It is sometimes an algebraic number

the procedure for solving when there are other variables present is the same as if everything is known other than the x.

Examples: solve the following for n

n – w = 9 an = 12 2k + n = 3k

5n = r w2n = t fn + 8 = b

more examples: solve for yax + y = 7 3x – hy = 12

5y + k = 3 – 2y 7(2w – my ) = 7w

Ch. 1 - section 2Data and scatter plots

Relation Definition: a statement that connects values of

one set of values of another set -

one set is designated the domain and the other set is designated the range

The relation is the statement (table, equation, graph) that determines which value in the range is connected to a given number in the domain

solutions are the pairs of numbers which satisfy the stated relation

Scatter plots - relations Data – information gathered in an effort to

determine in what way two sets are related – typically recorded in a table

Scatter plot – a graphical picture of the accumulated data

Key elements The domain and range are chosen – The choice is based on what relation you are

attempting to ascertain - in a table the domain is the first column and the range the second (third , fourth etc.) columns in a graph the domain is the horizontal axis and the range is the vertical axis independent variable – the variable chosen to represent the domain values

(typically x but can be any variable) Dependent variable – the variable chosen to represent the range values (typically y

but can be any variable) Solutions – ordered pairs containing a value from the domain and a value from the

range that SATISFY the relation

Scale – the units marked on the horizontal and vertical axis Label – the units (label) attached to the domain/range, independent/dependent

variables

“Eyeball” line of fit WHEN the data appears to ALMOST line up it is

determined that the best mathematical model will be a line.

A mathematical model is an equation that attempts to predict solution pairs beyond the data gathered.

In statistics you will learn how to find the line of BEST fit. For now we will “guestimate” this line by taking a ruler and attempting to draw a line down the center of the scattered dots.(or use the points that we are told to use)

Example: Reading a scatter plot

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Money in bank account

Day since paycheck $ in

hun

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Possible questions:

a. Estimate the amount of money in the bank on the third day after payday? Is the estimate the same as the actual amount? On the 10th day?

b. How many days did it take for the bank account to drop to $400

c. What is the domain?

d. What is the range?

Create a scatter plot Party attendance Given data – draw scatter plot A. determine independent and dependent variable B. determine a reasonable domain, range and scale C. Draw vertical and horizontal axis, LABEL them with scale and unit labels D. Plot individual points

girls Boys10 13

10 14

30 25

35 37

42 40

50 48

100 107

Ch1 – section 3Fundamentals of graphing/Slope and rate of change

General graphs unless you know something about an equation

creating its graph is tedious a graph is a picture of every solution pair – to create

a graph without any prior knowledge you make a table of solution pairs and put a dot down for each pair until you can see a pattern and connect the dots

y = x3 – 2x2 + 3. This is what the graphing calculator does. It is

therefore a useful tool when you don’t know anything about the equation.

Graphing linear equations Linear equations are named linear because it is

KNOWN that the pattern created from solution points will be a straight line

Example : graph y = 2x + 1 solutions are found: if x = 1 then y = 2(1) + 1= 3 so (1,3) x = 7 then y = 2(7) + 1 = 15 so (7,15) x = 3 then … Since You KNOW the pattern is a straight line 2 points are sufficient to draw the graph.

Using solution ponts to graph a line y = 5 – 2x

x y01

2

5 – 2(0)= 5

5 – 2(1)= 3

5 – 2(2) = 1

531

Example: Graph

x y0

3

6

3 +(2/3)(0)= 33 + (2/3)(3) = 53 + (2/3)(6) = 7

357

xy323

Example x= 3y – 5 While you can solve this for y, it isn’t necessary You can pick y values, get x values and graph

the points

Vertical and horizontal lines y = 5 x = 5

The geometry of a line Visually there are 4 categories of lines

Vertical

Increasing

Horizontal

decreasing

Slope Increasing and decreasing lines are described by the

angle which they form against the horizontal axis This angle can be measured by degrees (protractor)

or by slope - since getting out a protractor is inconvenient we tend to use slope.

slope is defined as a ratio between the vertical change (rise) and the horizontal change (run) of the line. Geometry guarantees that this ratio is constant for any place on the line for which it is measured -

rise

run

rise

run

𝑚=𝑟𝑖𝑠𝑒𝑟𝑢𝑛

𝑚=𝑟𝑖𝑠𝑒𝑟𝑢𝑛

𝑚=𝑚

𝒎=𝟐𝟑

Finding slope – given 2 points You could of course graph the 2 points and

count the slope Or you could understand the definition and the

mathematics slope is the ratio between rise (y’s) and run(x’s) m = (y2 – y1) / (x2 – x1)

examples: find slope (2,-4) and ( -6,12)

(3,7) and (6,8)

(-3,5) and (-5,10)

(8,4) and 8,-9)

( -3,7) and (2, 7)

Using slope to graph the line graph a line through (2,5) with a slope of 4/3

What slope tells you – (labels)

dollars

# of cows

Rate of change in a linear equation A rate of change is a ratio between the change in one object and the change in another object mi/hour words/page girls/boy miles per hour Words per page Girls for every boy

mathematics you should know – rate multiplied by denominator = numerator Thence the formula rt = d in a more general form r(b)= tALL linear equations have a number that is multiplying by the independent variable - This number IS a RATE OF CHANGE

Slope is the rate of change for the line that is graphed by the equation

The slope of the line that graphs the solutions of a linear equation is the same as the rate of change in the equation - slope is NOT x, it is m.

Slope intercept form rate of change and slope are the same number

(although not the same thing). In an equation, the rate of change is the number that you multiply x by to GET y.

Essentially this means solving for y and separating the fraction if there is one to see what number is multiplying by x

Slope is NOT an x or y co ordinate. It is not just the number in front of x.

Slope intercept FORM is not a number. It is the equation re written in a particular order.

Examples: put 2x + 3y = 12 into slope intercept form

Finding rate of change/slope y = 3x – 5 m =

y = 4 – 2x m =

2x + 5y = 10 m =

m =

3x – 4y = 12 m =

7x + 6y = 15 m =

Using slope to graph the line graph the line

Ch. 1 – section 4intercepts

Writing an equation in standard form

Ax+ By = C is call standard form of an equation where A and B are integers and A is not negative all linear equations can be written in standard

form – not all linear equations ARE IN standard form

Writing an equation in standard form is not solving – the goal is different – but it uses the same skills as solving

It is not a frequently used skill

Examples: write in standard form

Intercepts Def: Solution points that lie on the vertical or horizontal axis.

Thus you have vertical intercepts and horizontal intercepts – also referred to by the variable associated with the

range(dependent variable – generally y)(vertical) or domain(independent variable – generally x) (horizontal)

an intercept is a solution pair where ONE or both of the co-ordinates is 0.

Finding intercepts from graph

Finding intercepts from ALL equations(not just linear) the intercepts are points – solution- of the relation

therefore To find the vertical (y) intercept – solve/evaluate the

relation with the independent (x) variable = 0

To find the horizontal (x) intercept – solve/evaluate the relation with the dependent variable (y) = 0

Intercepts are convenient for graphing when the equation is in standard form

slope/intercept graphing is convenient when you are in slope/intercept form

Find the intercepts and graph the lines

RECALL : If the equation is linear then you KNOW that 2 points are sufficient for determining the direction of the line. Finding both the intercepts satisfies this need

-3x – 5y = 30

2w + 5v = 18 (write your answers as (v,w) )

y = 3x – 12

Significance of intercept While the vertical and horizontal intercepts are

frequently very important points in a given relation, their significance (meaning) is the same as any other point on the graph.

given any (x,y) – the significance is that for the value x the function yields the value y

Ex. A graph relates the number of hours you work to the pay(pounds) you receive.

Then the point (0, 10) means that for working 0 hours you will receive 10 pounds and the point (3,0) means that you work 3 hours and receive no pay.

Examples the relation between cost of pens (x) and cost of

notebooks(y) is given by the equation 3x + 5y = 15 Find the vertical (y) intercept and explain what it

means Find the horizontal (x) intercept and explain what

it means

Ch 1 – Section 5Writing linear equations

Finding the equation for a line Recap : given an equation for a line the line

can be determined from either 2 points that satisfy the equation or a description of the line (ie. Slope and one point- usually y-intercept)

Therefore given either 2 points or slope and one point you can determine a line. Since this information comes from the equation it is reasonable that given this information you can write the equation.

Method 1- find the m value and b value of the line and write

y = mx + b Examples: A. Write an equation for the line that goes through (0,7) and has a slope of -3

B. Write an equation for a line with a y-intercept of 3 and m= 0

C. Write an equation for a line with slope of 2/3 that goes through (9,2)

D. Write an equation for a line through (2,5) and (-1,7)

Method 1 (continued)Write equations for these lines

Method 2 : (y – y1) = m(x – x1) write an equation for the line that goes through (0,7)

and has a slope of -3

Write an equation for a line with a y-intercept of 3 and m= 0

Write an equation for a line with slope of 2/3 that goes through (9,2)

Write an equation for a line through (2,5) and (-1,7)

Method 2 (continued)

Parallel and perpendicular lines

FACT: parallel lines have the same slope FACT: perpendicular lines have slopes that are

both opposite signs and reciprocal (ie: if you multiply the slopes of perpendicular lines the product is -1)

Determine if 2 lines are parallel, perpendicular or neither y = 2x – 7 y = 7 + 2x

y – 5x = 3

Ch 1 – Section 6Linear models

Linear models from data A linear model is an equation that is used to predict

values of y and x based on known information A linear model is a linear equation where x and y

have labels associated with them

To find a linear model determine pertinent information from the table or words used to describe the data. – pertinent information is slope(rate of change) and related numbers (ordered pairs)

Example: An ant is traveling 3 feet every 2 seconds in a

straight line towards a food spill. After 3 seconds he is 5 feet away from the spill.

Write a linear model for this situation. When will the ant arrive at the food spill? How far away from the spill was he when we

started timing his progress? Where was he 3 sec. before we started timing his

progress?

Linear models from scatter plots

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Money in bank account

Day since paycheck $ in

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Write a linear model for the money in this bank account on any given day.

Ch 1 – Section 7Functions and function notation

Definition a relation is any rule (table, graph, equation, or procedure)

that connects numbers in a domain to numbers in the range. Domain vs range is frequently an arbitrary designation

A function is a relation in which the domain is designated and each value of the domain has a unique output from the range

ie : we say that “y is a function of x” if for any given value of x(domain – input) you can find only one value of y(range – output) which satisfies the given “rule” (table, graph, equation, words)

Examples: determine if the following are functions Procedures (a rule stated in words) tables (finite list of points – or partial list of

infinite set showing a pattern) Graphs (on the coordinate plane) Equations (two variables implied)

proceduresfunction Match each student id to

the students gpa

Not function Match each gpa to the

student who received that grade

Tablesfunctionx -3 -1 1 3y 2 3 2 3

Not functionx 1 1 3 4y 1 2 3 4

Graphfunction Not function

equationsFunction Y = 3x + 5

Not function x = 7

Notation : f(x) = y (read: f of x equals y) means – there is a rule called f for which you input

the value of x to arrive at a value or equation that is equal to y (Clark considers the output variable to be the same as the name of the function: f(x) = f )

f can be a table, graph, equation, or procedure x can be a numeral, variable, variable expression

or another relation y is a numeral or variable expression

Examples: What does the function notation mean? Given : v(x) gives volume in cu. in. at a given celcius

temperature

The input variable (called independent variable) is ___, the output variable (called dependent variable) can be any that you desire.(Clark wants you to choose ____)

Thus v(68) = 40 means?

v(x) = 92 means?

v(78)= y means?

Answering questions using a function statement- example Given that c(u) = 30u + 50 is a function for the

cost of producing a given number of items c(u) = 30u + 50 means that when I use a “u” in

the function rule I get the formula 30u + 50.

Find c(10) and explain what it means.

Find c(u)= 200 and explain what it means.

Another example k(x) = 7 - x is a function.

Find k(9) find k(x) = 9

Using a table to answer f(x) questions

given the table below. find v(7) v(5) v(-3) ??? v(8)

find for what values v(x) = 2 v(x) = -9 v(x)=7x -7 -5 -3 -1 1 3 5 7 9

v(x) -6 3 -9 2 -4 3 6 -12 3

Using graphs to answer f(x) Estimate m(2) m(-4) m(x) = -4

f(x)=7-(2/3)x

x

y

m(x)