Expressions, Rules, and Graphing Linear Equations by Lauren McCluskey This power point could not...

Expressions, Rules, and Graphing Linear Equations by Lauren McCluskey

• This power point could not have been made without Monica Yuskaitis, whose power point, “Algebra I” formed much of the introduction.

Variable:

• Variable – A variable is a letter or symbol that represents a number (unknown quantity or quantities).

• A variable may be any letter in the alphabet.

• 8 + n = 12“Algebra I” by M. Yuskaitis

Algebraic Expression:

• Algebraic expression – a group of numbers, symbols, and variables that express an operation or a series of operations with no equal or inequality sign.

• There is no way to know what quantity or quantities these variables represent.

• m + 8• r – 3

“Algebra I” by M. Yuskaitis

Simplify

• Simplify – Combine like terms and complete all operations

m = 2

• m + 8 + m 2 m + 8

• 3x + (-15) -2x + 5 x -10

“Algebra I” by M. Yuskaitis

Evaluate

• Evaluate an algebraic expression – To find the value of an algebraic expression by substituting numbers for variables.

• m + 8 m = 2 2 + 8 = 10

• r – 3 r = 5 5 – 3 = 2

“Algebra I” by M. Yuskaitis

Translating Words to Algebraic Expressions• Sum Difference

• More than Less than

• Plus Minus

• Increased Decreased

• Altogether

“Algebra I” by M. Yuskaitis

Translate these Phrases to Algebraic Expressions

• Ten more than a number

• A number decrease by 4

• 6 less than a number

• A number increased by 8

• The sum of a number & 9

• 4 more than a number

n + 10n - 4

x - 6n + 8n + 9

y + 4“Algebra I” by M. Yuskaitis

Each of these Algebraic Expressions might represent Patterns:• For example: n + 10

(x) (y)

1 11

2 12

3 13

Or it might be Geometric (n-4):

n=1

13 seats

n=2 8 seats

Patterns

Patterns may be seen in:

• Geometric Figures

• Numbers in Tables

• Numbers in Real-life Situations

• Sequences of Numbers

• Linear Graphs

Patterns are predictable.

Patterns with Geometric Figures (Triangles)

• Jian made some designs using equilateral triangles. He noticed that as he added new triangles, there was a relationship between n, the number of triangles, and p, the outer perimeter of the design. Write a rule for this pattern.

from the MCAS

P=3

P= 4

P=5

P=6

How to Write a Rule:

1) Make a table.

2) Find the constant difference.

3) Multiply the constant difference by the term number (x).

4) Add or subtract some number in order to get y.

1 ) Make a Table: Let x be the position in the pattern while y is

the total perimeter. # of Triangles Rule: Perimeter

(x) (y)

1 ? 3 2 4 3 5 ... … x y

from the MCAS

P = 3

P = 4

P = 5

P = 6

2)Find the Constant Difference: How did the output change?

Perimeter (y)

3 4 5 6 … p from the MCAS

+1

+1

+1

P = 3

P = 4

P = 5

P = 6

3) Multiply by the Input # (x). 4) Then Add or Subtract some # to get the Output # (y).

# of Triangles Rule: Perimeter

(x) (y) 1 1x +2 3 2 1x +2 4 3 1x +2 5 ... … x y

It Works! from the MCAS

P = 3

P = 4

P = 5

P = 6

Patterns in Numbers in Tables:

• Write a rule for the table below.

Input (x) 2 5 10 11

Output (y) 5 11 21 23

from the MCAS

2) Look for the Constant Difference.Input (x) 2 5 10 11

Output (y) 5 11 21 23

•What is the change when the input # increases by 1? •From the 10th to the 11th the output #s increase from 21 to 23.

So the constant difference is +2.

3) Multiply x by the Constant Difference. Then… 4) Add or Subtract some #.

Input (x) 2 5 10 11

Output (y) 5 11 21 23

2Constant Difference

Input #

Constant

+1x

Patterns in Numbers in Real-Life

Situations:

from the MCAS

Write a rule for x number of rides:

1) Make a Table:

In (x)

# of Rides

Out (y)

Cost

1 $

2 $

3 $

12

14

16

2) Find the Constant Difference.

In (x)

# of Rides

Out (y)

Cost

1 $12

2 $14

3 $16

+$2…

+$2

+$2

So the Constant Difference is +2.

3) Multiply x by the Constant Difference.Then…4) Add or Subtract some #. In (x)

# of Rides

Out (y)

Cost

1 $12

2 $14

3 $16

Constant Difference

Input # Constant

2 x +10

Patterns in Sequences of Numbers

Remember:

1) Make a Table.

2) Find the Constant Difference.

3) Multiply x by the Constant Difference.

4) Add or Subtract some #.

12, 16, 20, 24…

What’s my rule?

1) Make a Table:

(x) (y)

1 12

2 16

3 20

+4

+4

2) Find the Constant Difference.

The Constant Difference is +4.

3) Multiply x by the Constant Difference. Then…4) Add or Subtract some #.

(x) (y)

1 12

2 16

3 20

Constant Difference

Input # Constant

4 x +8

Patterns in Linear Graphs

Remember: 1. Make a Table.2. Find the Constant Difference. 3. Multiply x by the Constant Difference.4. Add or Subtract some #.

“Linear” means it makes a straight line.

To Make a Table from a Graph: (x) (y)

-1 -3

0 -1

1 1

Find the Constant Difference.

+2

+2

3) Multiply x by the Constant Difference. Then…4) Add or Subtract some #.

(x) (y)

-1 -3

0 -1

1 1

Constant Difference

Constant

2 x -1

Input

How to find the 10th or 100th term:

• Now that we have a rule we can find any term we want by evaluating for that term #.

• Just substitute the term number for x, then simplify.

What would ‘y’ be if x = 10?

The rule for the last graph was: 2x -1

Substitute 10 for x and we get: (2)(10) – 1 or 20 -1 = 19. So (10, 19) are

solutions for this rule,

AND (10, 19) would be a point on this line!

What would ‘y’ be if x = 100?

2x – 1 was the rule for the graph. Substitute 100 for x:(2)(100) – 1 or 200 -1 = 199 So (100, 199) would be a solution for this

rule,

AND (100, 199) would be on this line!

ReviewSo here we have come full circle, we have: Written algebraic expressions; Evaluated these expressions;Written expressions (rules) for patterns; Evaluated these rules for specific terms.

Graphing Linear Patterns

There are 3 forms of equations

that can be graphed:

1) Slope-intercept form

2) Standard form

3) Point-slope form

Slope-Intercept Form (Slope)

• The “slope” of a line is the measure of its steepness.

rise

run

Or: Rise overRun

Y-Intercept:

• The y-intercept is the point where a line crosses the y-axis.

• Hint: Think of the word, ‘intersection’, where 2 streets cross, in order to remember ‘intercept’.

-1

Finding the Slope on a Graph:The slope of the line is rise run.Or: the change in y the change in x.

Change in y = 2 2Change in x = 1 1

So the slope is +2.

=

Kinds of Slopes:

•Slopes may be positive (y increases as x increases);•Slopes may be negative (y decreases as x increases);•Slopes may be zero (y doesn’t change at all); •Or Slopes may be undefined (x doesn’t change at all).

Name the Type of Slope:

Slope-Intercept Form:

slope

y-intercept

2 x -1

You can see both the slope and the y-intercept on the graph:

Standard Form: • It’s easy to find the x- and y-intercept

with the standard form (Ax + By = C).

• All you need to do is substitute “0” for x and solve for y; then substitute “0” for y and solve for x.

Try it:

Write y = 2x -1 in standard form:y = 2x - 1 -2x -2xy - 2x = -1

y - (2) (0) = -1 y = -1

So the y-intercept is -1.

0 - (2) x = -1

-2 -2 x = 1/2

So the x-intercept is 0.5.

Point-Slope Form: The point slope form (y - y1) = m(x - x1) is easiest to use if you are given one point and the slope of the line.

Just substitute the coordinates into the equation. Then rewrite the equation in slope-intercept form.

Point-Slope Form

•Suppose you did not have the graph, but you were told that the point (2, 3) is on the line and the slope is +2…

•You could write the equation: y - 3 = 2(x - 2),then rewrite it in slope-intercept form.

Point-Slope Form:You could rewrite y - 3 = 2(x - 2) to the slope-intercept form:

y - 3 = 2(x - 2)y - 3= 2x - 4 +3 +3y = 2x -1

Slopes of Parallel Lines:

Two lines on the same plane that have the same slope will be parallel.

Slope is 0.

Slope is undefined.

Slopes of Perpendicular Lines:

Two lines whose slopes are negative reciprocals are perpendicular. The product of their slopes will equal -1.

Note: Perpendicular lines form right angles at their intersection.

Are they Parallel or Perpendicular?

y = 2x + 10 y = 2x -5

y = -3x + 2y = 1/3x + 1