Clark/Anfinson. Linear Functions Solving/Evaluating - linear equations.

Clark/Anfinson

Linear Functions

Solving/Evaluating - linear equations

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

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

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

e. 7(3x 2) = 14 f. 4q 12 + 3q = -47 g. 2 (x 9) + 3x = 2

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)

Method 1 Method 2

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.

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

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?

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.

n w = 9 an = 12 2k + n = 3k 5n = r w 2 n = t fn + 8 = b

ax + y = 7 3x hy = 12 5y + k = 3 2y 7(2w my ) = 7w

Data and scatter plots

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

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

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

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)

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 10 th 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?

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 girlsBoys 1013 1014 3025 3537 4240 5048 100107

Fundamentals of graphing/Slope and rate of change

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 = x 3 2x 2 + 3. This is what the graphing calculator does. It is therefore a useful tool when you dont know anything about the equation.

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.

y = 5 2x xy 0 1 2 5 2(0)= 5 5 2(1)= 3 5 2(2) = 1 5 3 1

xy 0 3 6 3 +(2/3)(0)= 3 3 + (2/3)(3) = 5 3 + (2/3)(6) = 7 3 5 7

x= 3y 5 While you can solve this for y, it isnt necessary You can pick y values, get x values and graph the points

y = 5 x = 5

Visually there are 4 categories of lines Vertical Increasing Horizontal decreasing

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

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 (ys) and run(xs) m = (y 2 y 1 ) / (x 2 x 1 )

(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)

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

dollars # of cows

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)= t ALL 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.

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

y = 3x 5 m = y = 4 2x m = 2x + 5y = 10 m =

intercepts

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

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.

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

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.

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

Writing linear equations

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.

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)

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)

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)

Linear models

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)

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?

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

Functions and function notation

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)

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)

function Match each student id to the students gpa Not function Match each gpa to the student who received that grade

function x-313 y2323 Not function x1134 y1234

functionNot function

Function Y = 3x + 5 Not function x = 7

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

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?

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.

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)=7 x-7-5-313579 v(x)-63-92-436-123

Estimate m(2) m(-4) m(x) = -4 m(x)

Clark/Anfinson. Linear Functions Solving/Evaluating - linear equations.

Documents

Transcript of Clark/Anfinson. Linear Functions Solving/Evaluating - linear equations.