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UNIT 4 NOTES

4-1 and 4-2 Coordinate Plane

Ordered pairs on a graph have several names.

(X coordinate, Y coordinate)

(Domain, Range)

(Input,Output)

Plot these points and label them:

a. (3,-4) b. (-5,2) c. (0,4)

d. (2,0) e. (-4,-3)

A function is where each x coordinate is paired with exactly one y coordinate.

The X VALUES CANNOT REPEAT, but the Y VALUES CAN.

EX: Is the relation a function? {(2,1), (3,1), (5,2)} Domain

___________Range__________

Why?______________________________________________________

EX: Is the relation a function? {(1,2), (3,1), (1,5)} Domain

___________Range__________

Why? ______________________________________________________

The VERTICAL LINE TEST tells you whether a relation is a FUNCTION if the graph hits a vertical

line (your pencil) at only ONE POINT (in other words, the X value does not repeat!)

0 x

y

0 x

y

0 x

y

2

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4-4 Writing a Function Rule

You can write a rule for a function by

analyzing a table of values.

Find the pattern!

Write a function rule for each table of

values.

1.

x f(x)

2 3

4 5

6 7

8 9

2.

x f(x)

-3 3

0 0

3 -3

6 -6

4

3.

x f(x)

0 -2

2 0

-2 -4

4 2

Write a function rule to describe each

statement.

4. the amount of money you earn mowing

lawns m(n) at $15 per lawn

5. the cost in dollars of printing dollar bills

c(d) when it costs 3.8 cents to print a dollar

bill.

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4-5 Slope

Slope (m) - The steepness of a line

Positive Slope

Negative Slope

Zero Slope

No Slope (undefined)

Uphill

Downhill

Flat

(horizontal)

Vertical

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Ways to find slope

Graph: Rise

Run

2 PointsFormula:

1 1 2 2( , )( , )x y x y

2 1

2 1

y ym

x x

1. Find the slope of:

2.

Find the slope of the line that passes through:

3. (–4, –2) and (4, 4)

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4. (21

2, −1

1

2) and (−

1

2,

1

2)

5. (–5, –3) and (–5, 1)

6. (–7, 4) and (2, 4)

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4-6 Slope Intercept Form

y-intercept: where the graph crosses

the y-axis

Slope Intercept Form:

y = mx + b

where m is the slope &

b is the y-intercept

Find the slope &

y-intercept of each:

1. 3 1y x

2. 2

73

y x

3. 2

3y x

4. y = 5

9

5. 3 2 9y x

6. 1

4 23

x y

Write an equation for each line:

7. 1

4m and b = –10

8.

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Ordered Pair Solution

A point lies on the line of a graph if it is

a solution to the equation.

Plug in the x- & y-coordinates

Use the slope & y-intercept to graph each line:

9. 1

42

y x

10. 5 2y x

11. Does (–3, 4) lie on the graph of y = –2x + 1?

(1, –1)?

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4-7 Standard Form

Standard Form:

Ax + By = C

where A, B, and C are integers

(so they can’t be decimals or fractions)

x-intercept: where the graph crosses the

x-axis

To find the x-intercept make y = 0

y-intercept: where the graph crosses the

y-axis

To find the y-intercept make x = 0

x-int y-int

y = 0 x = 0

Ways to graph linear Equations:

- x- and y-ints

- slope & y-int

Write in standard form:

1. 3 2y x

2. 1

52

y x

3. Find the x- and y-ints of: 5x – 3y = –12

x-int y-int

y = 0 x = 0

4. 2 5 10x y

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Special Equations:

x =

-no y in the equation

-vertical line no slope

y =

-no x in the equation

-horizontal line m = 0

5. 4 1y x

6. 3 2 6x y

7. 2x

8. 2y

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4-8 Point-Slope Form and Writing Linear Equations

Point-Slope Form:

1 1y y m x x

Where m is the slope & 1 1,x y is a point.

Graph:

1. y - 2 = 2 (x-3)

2. 5 2y x

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Finding an equation when given two points:

1. Find the slope

2. Use one of the points & the slope

point slope form

3. Rewrite Equation

Write an equation in point-slope form:

3. (2, –5) m = 1

2

4. (–5, 6) m = undefined

Write an equation for the line in point slope &

slope-intercept form:

4. (3, 5) & (0, 0)

5. (–3, 12) & (6, 1)

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4-9 Point-Slope Form and Writing Linear Equations

For each pair of points find the:

a. Slope d. Equation in Standard Form b. Equation in Point-Slope Form e. x– and y– intercepts c. Equation in Slope-Intercept Form f. Graph of the line

1. (1, 5) & (–4, 2)

2. (2, –1) & (8, –4)

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3. (2, 7) & (2, –4)

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4-10 Parallel & Perpendicular Lines (part 1)

Parallel Lines:

-never intersect

-have the same slope

Ex: m = 4 & m = 4

Perpendicular Lines:

-Intersect at right angles (90˚)

-Slopes are negative reciprocals

Ex: m = ½ & m = –2

Find the slope of the line that is parallel and

perpendicular to each line.

1. 3 2y x 2. 3 4 8x y

3. 7x

Determine whether the graphs of each pair of equations

are parallel, perpendicular or neither.

4. 𝑥 − 2𝑦 = −4 2𝑥 + 𝑦 = −2

5. 3𝑥 + 𝑦 = 3

6𝑥 + 2𝑦 = −10

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4-10 Parallel & Perpendicular Lines (part 2)

Finding Parallel or Perpendicular

Lines

1. Find the slope of the

original equation

2. Find the

parallel/perpendicular slope

3. Use the point & new slope

in point-slope form

4. Change to standard form

Write an equation in slope-intercept form of the line that passes

through the given point & is parallel to the graph of each equation:

1. 5𝑥 + 𝑦 = 2, (2,3)

2. 3𝑥 − 2𝑦 = 7, (−3,1)

Write an equation in slope-intercept form of the line that passes

through the given point & is perpendicular to the graph of each

equation:

3. 7𝑥 − 2𝑦 = 3, (4, −1)

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4. 2𝑥 + 5𝑦 = −3, (2, −3)

4-11 Fitting Equations to Data

Steps to fitting equations to data.

1) Identify two points

2) Find slope

3) Use a point and slope to write a

linear equation. Use point-slope

formula.

4) Solve for the unknown by

substitution.

1. To produce 50 copies of a school newspaper, the cost per paper is

26 cents. To produce 200 newspapers, the cost per paper is 20 cents.

Let n be the number of copies of a school newspaper, and let c be the

cost per paper. Assume that a linear relationship fits these data with

ordered pairs (n, c)

(1) Find the linear equation that fits these data.

(2) Use the linear equation to predict what it would cost per paper

to produce 300 copies.

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(2) To find the cost per paper for 300 papers, substitute 300 for n and

solve for c.

2. A college record in the 100-m dash in 1960(t) was 10.5 seconds(r).

In 1990 the new record was 10.2 seconds. Assume a linear

relationship fit these data with ordered pairs (t, r).

(1) Find a linear equation to fit the data points.

(2) Use the linear equation to predict the record in 2020