Section 4 1: Slope

Section 4 – 1: Slope

Which line is steeper?

Slope:

1)  The ratio of the change in the y-coordinates (______) to the change in the x-coordinates (______).

2)  Notated with the letter m.

3)  Slope is:

Examples:

State the slope of the following line.

1). 2).

3). 4).

Finding the Slope of Two Points.

The slope m of a nonvertical line through any points and can be found as follows

Example: State the Slope of the line

1)  The line that goes through (-3, 2) and (5, 4).

2)  The line that goes through (-3, 4) and (4,4).

3)  The line that goes through (-3, 4) and (-2, 8).

4)  The line that goes through (-2, -4) and (-2, 3).

Concept Summary: Classifying Lines

Positive Slope / Negative Slope / Slope of Zero / Undefined Slope

Example:

Find the Error: Kate and Dave are finding the slope of the line that passes

through (2,6) and (5,3).

Kate: Dave:

Who is correct?

4-1 Slope as a Rate of Change

Slope can be interpreted as the rate of change of a line. This means that slope tells us how y changes as x increases. The change in y can be an increase (positive slope) or a decrease (negative slope).

Example 1

The slope of this line is _____.

This means that y (increases or decreases)

____ unit(s) for every _____ unit(s) that x

increases.

Example 2

The slope of this line is _____.

This means that y (increases or decreases) ____ unit(s)

for every _____ unit(s) that x increases.

A graph may represent a real-world situation. The slope, or rate of change, will have units of measurement that allow us to interpret the rate of change.

Example 1

The graph shows the distance traveled in miles

given the amount of time traveled in hours.

1) Write the slope as a rate of change.

2) Interpret the slope in the context of the problem.

Example 2

The graph shows the amount of water in

gallons that flows through two different

pipes after a certain number of minutes.

1) What is the flow rate of pipe A?

2) What is the flow rate of pipe B?

3) The flow rate of pipe A is greater than the flow rate of pipe B. How is that evident from the graph?

Example 3

1) Explain what information the graph shows.

2) From 0 minutes to 400 minutes, the graph has a slope of 0.

Explain what this means in the context of the problem.

Example 4:

Naomi left from an elevation of 7400 feet at 7:00am and hiked to an elevation of 9800 feet by 11:00am. What was her rate of change in altitude?

Section 4 - 4 Writing Equations in Slope Intercept Form

Remember:

Slope intercept form: ______

m =

b =

Example 1)

Find the linear equation given the slope and y-intercept.

1). Slope = -4 and y-intercept = 3 2). Slope = and y-intercept = -7

Example 2)

Find the linear equations given a graph.

1) 2)

Steps: 1)

2)

3)

Example 3)

Writing linear equations give the slope and a point.

1) The slope is 3 and ordered pair (1, 4).

What is m =

What is x =

What is y =

Substitute into y = mx + b and solve for b.

2) The slope is and ordered pair (-4, 10).

3) The slope is and ordered pair (5, 12).

4) The slope is -3 and ordered pair .

Example 4)

Writing linear equation given two points.

1) Given the ordered pairs ( 6, -6) and ( -4, -4).

Find the slope first m =

Pick one of the ordered pair to substitute into y = mx + b

Solve for b

2) Given the ordered pair ( -5, 4) and ( -5, -1).

3) Given the ordered pair ( -2, 0) and ( 1, -1).

4) Given the ordered pair ( -2, 3) and ( 8, 3).

Section 4 – 3: Graphing Equations in Slope Intercept Form

Slope-Intercept Form:

·  m represents the slope of the line

·  b represents the y-intercept of the line.

y – intercept (b) = is the point where the graph crosses the y – axis at the

point ( 0, y).

Example:

State the slope and y – intercept.

1) 2) 3.)

Write Equation in Slope Intercept Form

1.) 2.)

3.) 4.)

Write and Equation from the Line and Points:

Write an Equation from the Line:

Graphing Linear Equations

Graph the following equations.

1)

2)

3)

4)

5)

6)

Section 4.5 Writing Equations in Point-Slope Form

Point – Slope Form:

is from given point

m is the slope of the line.

Example 1)

Write the point-slope form of an equation for the line that passes through (-1, 5) with the slope of (-3).

Example 2)

Write y – 2 = 3(x + 5) in slope intercept form

Example 3)

Use the point slope formula to find the linear equations given two points.

***(Hint: First Find Slope)***

1) Given the ordered pairs ( 5, 1) and ( 8, -2)

2) Give the ordered pairs ( 6, 0) and ( 0, 4).

****Practice: Pg. 223 in Book

Section 4 -6 Line of Best Fit

Defintions:

Scatter Plot –

Line of Fit –

Best-Fit Line –

Example:

Negative Correlation Post Correlation

Is this Correlation Positive or Negative?

a. b.

Example 1.)

-  The Table shows the Largest vertical drops of nine roller coasters in the United States and the number of years after 1988 that they were opened.

Years Since 1988 / 1 / 3 / 5 / 8 / 12 / 12 / 12 / 13 / 15
Vertical Drop (ft) / 151 / 155 / 225 / 230 / 306 / 300 / 255 / 255 / 400

a. Draw a scatter plot and determine what relationship exists, if any, in the data.

b. Draw a line of Fit

c. Write the slop intercept form of the equation

*** Using your equation what should the largest vertical drop be in 2014??***

Section 4 – 7 Parallel and Perpendicular Lines

Parallel Lines: Lines in the same plane that do not intersect.

*****Parallel Lines have the same Slope.******

Example:

y = 2x + 3 & y = 2x -7

Write an equation for a line, given a point and the equation of a line parallel to it.

(Use point slope form, then write in slope intercept form)

1. The point (-1,-2) and parallel to y = -3x -2

2. The point (3,5) and parallel to 2x + 4y = 12 (solve for y first)

3. The point (-2,4) and parallel to y = 3

4. The point (-3,-5) and parallel to x = 1

Perpendicular Lines: Lines that intersect at right angles.

***Perpendicular Lines are Opposite Reciprocals*** ex: 4 = -1/4

Example:

y = 5/3x + 4 and y = -3/5x + 2

Write an equation for a line, given a point and the equation of a line perpendicular to it

1. The point (-3,-2) and perpendicular to graph of y = ¼x + 3

2. The point (1,6) and perpendicular to graph of 4x – 2y = 6 (solve for y)

3. The point (5,-2) and perpendicular to graph of y = 3

4. The point (-4,0) and perpendicular to graph of x = -1

Determine whether two lines are perpendicular, parallel or neither

1.  y = x +3 and y = -3/4x -2

2. y = x + 5 and y = -2x + 3

3. y = 2x + 6 and -4x + 2y = 20

4. 4x + 8y = 2 and -x – 2y = 6

Concept Summary Forms of Linear Equations

Form / Equation / Description
Slope-Intercept
Point-Slope

Example:

A line has a slope -2 and passes through the point (5, 3). Write the equation of the line in:

a)  Point-Slope form:

b)  Slope-Intercept form: