Name: ____________________________Date: ____________________________

Linear Relations- Student Notes and Guided Examples

Date: 1 – Writing Equations to Describe Patterns _______________2 – Linear Relations _______________3 – Another Form of the Equation for a Linear Relation _______________4 – Matching Equations and Graphs _______________5 – Using Graphs to Estimate Values _______________

What You’ll Learn:

1 – Use expressions and equations to generalize patterns2 – Verify a pattern by using substitution3– Graph and analyze linear relations4 – Interpolate and extrapolate to solve problems

One definition of math is ‘the study of patterns’. How can finding and studying patterns be important in our world?

What word is inside the word ‘linear’? So what may a linear relation be? Can you think of a real world example of a linear relation?

_____________1 – Writing Equations to Describe Patterns_______________________Focus: Use equations to describe and solve problems involving patterns.Try the following ‘Investigate Activity’:

Warmup: Look at the ‘investigate’above. Can youdescribethe pattern:a) in wordsb) a tablec) an equation where t = number of tables & p = number of peopled) How many people at 25 tables?

Ex1You go to the storeto buy peanut butter.a) Make notes on oraround the table.b) Write an equationthat relates the cost tothe number of jars of

Tables People1 423

Jars Cost ($)0 01 52 103 154 20

peanut butter.c) What is the cost of 18 jars?

Ex2Find the equationusing the pattern inthe table

Ex3Find the equationusing the pattern inthe table

If a = 26, what is b?

Ex4Empress Cabs charges$3.75 plus $1.25 per km.a) if d = number of km,write an equation for thecost of Empress Cabs.b) What is the fare for a 19 km ride?

Reflection: What is the biggest challenge for you in this section and how do you think you can improve upon it?

__________________________2 – Linear Relations______________________________

m n0 11 32 53 7

a b3 -16 09 112 20 -2

Focus: Analyze the graph of a linear relation.Main Ideas:

Warmup: The x axis is _________________ and the y axis is ______________ How do you plot points The point will be given as, for example, (3, -5). The first number on a coordinate plane? is always the x value and the second is the y value. Start at (0, 0) on the graph (the middle). If x is positive, go ________. If x is negative go _________. From there, count your y. If y is

positive, go _____, and if negative, go ________. Then plot your point.

Plot and label the points on the coordinate plane. a) (0, 0) b) (5, 0)c) (0, 3) d) (-4, 0)e) (3, 7) f) (-8, 5)g) (0, -6) h) (-4, -7)

Ex1Suppose you were monitoring daily temperature. Three days ago, the temperature was -7C.Everyday since, the temperature has/will increase by 3Ca) Complete the table.b) Graph the relation.c) What kind of patternand/or relationshipsdo you notice in the table and/or graph?Ex2

Day (x axis) Temp (y)-3 -7-2-10123

On a table, x is always listed first, then y.

DVDs Cost ($)

1 2.502 5.003 7.504 10.00

The table shows the cost of renting DVDsat an online store. a) Graph the points, butdon’t draw a line.b) Use the table to describe the pattern inthe rental costs. How isthis pattern shown in thegraph?c) Why don’t we drawa line?

Is the number of DVDspurchased related to thecost?

What is the equationfor DVDs and costin example 2? Use x forDVDs and y for cost.

What does the equationtell us?

Does cost depend on thenumber of DVDs, ordoes the number of DVDs depend on cost?

What is an independentvariable, and what is adependent variable?

What is a linear relation?

A relation has the

x y-2

equation y = 5 – 2x. a) Create a table of values for values of x from -2 to 4. Find y foreach.b) Graph the relation.Should you join the points with a line?c) What patterns do yousee in the table and graph?d) Is the relation linear?

If you know a relationis linear, how many points do you need toplot the line?

Reflection: If you were to plot a linear relation between number of km driven vs. cost of gas, which of the two would be the independent variable and which would be the dependent variable. Explain.

___________3 – Another Form of the Equation for a Linear Relation______________

Focus: Recognize the equations of horizontal, vertical, and oblique lines, and graph them.Main Ideas:

Warmup: Suppose you have a piece of licorice 10cmlong.a) How many differentways could you cut itinto two pieces?b) In words, how are the lengths of the twopieces related?c) If x = the length of the first piece, and y =the length of the secondpiece, write an equationfor the relation.d) How is your equationdifferent from the equations we workedwith in 4.2?e) Make a table of values.f) Graph the equation.g) Is the relation linear?

Ex1For the equation3x – 2y = 6:a) Make a table of values for x = -4, 0, 4b) Graph the equation.c) What is another namefor a ‘slanted’ line?

Piece #1 (x)

Piece #2 (y)

Sometimes, only onevariable appears in anequation, for example, x = 2, or, y = -5

Ex2Graph x = 2Hint: The only requirement for a pointon the graph is that thex value must be 2. Soy can be anything, as long as x is 2.

What kind of line isproduced?

Ex3Graph x = -1

Ex4Graph y = -5 The only requirement isthat for each point, they value must be -5.

What kind of line isproduced?

What kind of equationProduces a:a) oblique line?b) vertical line?c) horizontal line?

Reflection: Give an example of an equation that produces a: (i) horizontal line (ii) oblique line (iii) vertical line

__________________4 – Matching Equations and Graphs________________________Focus: Match equations and graphs of linear relations.

Investigate:

Warmup:

Do the ‘investigate’. Show any worketc. on the right.

Look over the ‘Connect’on p. 184.Let’s look at the best wayto match an equation withits graph.How do you choose xvalues to test?

Ex1 Why is testing x = 0 unhelpful on the graph to the right?

Ex 2

When all lines cross they axis at a different point, why is it smartto test x = 0 into theequation(s)?

__________________5 – Using Graphs to Estimate Values________________________Focus: Use interpolation and extrapolation to estimate values on a graph.Investigate:

Warmup: Do the ‘Investigate’.

What is interpolation?

When did you use interpolation in the Investigate?

What is extrapolationWhen did you use extrapolation in the

investigate

Look at the graphsAbove to better understandextrapolation.

Ex1Corey goes biking. Every 3 minutes (x), Corey travels 1.5km (y). a) Draw a graph for thefirst 12 minutes of biking, but leave room at the end of the graph.

b) How far has Coreybiked after 7 minutes?Is this interpolation orextrapolation?c) How far will Coreyhave biked after 17minutes. Extend yourgraph to 18 minutes toassist you. Is this interpolation or extrapolation?

Reflection: Give one real life example of extrapolationthat could have a significant impact on our world.

Ex 2- Do the following question by first re-drawing andextending the graph.Do part a (i) & (iii)and part b (i) & (iii)