Unit 4 - Linear Equasions

Slope, Slope intercept, Point Slope, Parallel and Perpendicular Lines, Correlation Lines of Fit, and Residuals.

Slope

The steepness of a line measured by the change in y divided by the change in x

m = (y₂ − y₁) / (x₂ − x₁)

What is the formula for slope using two points?

The line rises from left to right

What does a positive slope tell you about a line?

The line falls from left to right

What does a negative slope tell you about a line?

0

What is the slope of a horizontal line?

Undefined

What is the slope of a vertical line?

The slope

In the equation y = mx + b what does m represent?

Divide the change in y by the change in x to get a slope of 2

How do you find the slope if x increases by 3 and y increases by 6?

The equation of a line written as y = mx + b

What is slope-intercept form?

The slope

In y = mx + b what does m represent?

The y-intercept

In y = mx + b what does b represent?

The point where a line crosses the y-axis

What is the y-intercept?

The slope is 4 and the y-intercept is −2

What are the slope and y-intercept in y = 4x − 2?

Start at the y-intercept then use the slope to find another point

How do you graph an equation in slope-intercept form?

−3

What is the slope of y = −3x + 5?

No because vertical lines cannot be written in the form y = mx + b

Can every line be written in slope-intercept form?

The equation of a line written as y − y₁ = m(x − x₁)

What is point-slope form?

y − y₁ = m(x − x₁)

What is the formula for point-slope form?

The slope of the line

In point-slope form what does m represent?

A point that lies on the line

In point-slope form what does the point (x₁ y₁) represent?

When you know the slope and one point on the line

When should you use point-slope form?

y − 5 = 3(x − 2)

What is the point-slope equation for a line with slope 3 through (2 5)?

y − 7 = −1(x − 4)

What is the point-slope equation for a line with slope −1 through (4 7)?

Solve the equation for y

How can you rewrite point-slope form as slope-intercept form?

The matching x-coordinate and y-coordinate of the known point

What should you substitute for x₁ and y₁ in point-slope form?

They are equal

What are the slopes of two nonvertical parallel lines?

They never intersect

What do parallel lines in the same plane do?

A right angle

What angle do perpendicular lines form?

They are negative reciprocals

What is the relationship between the slopes of two nonvertical perpendicular lines?

−1/2

What is the negative reciprocal of 2?

Yes because they have the same slope

Are the lines y = 3x + 1 and y = 3x − 4 parallel?

Yes because their slopes are negative reciprocals

Are lines with slopes 4 and −1/4 perpendicular?

A vertical line

What kind of line is perpendicular to a horizontal line?

The relationship between two variables

What does correlation describe?

Both variables tend to increase together

What does a positive correlation mean?

One variable tends to decrease as the other increases

What does a negative correlation mean?

There is no clear linear relationship between the variables

What does no correlation mean?

A scatter plot

What kind of graph can show correlation?

The points lie close to a clear pattern

What does a strong correlation look like on a scatter plot?

The points are spread widely around the pattern

What does a weak correlation look like on a scatter plot?

No correlation alone does not prove causation

Does correlation prove that one variable causes another?

A line that models the overall trend in a scatter plot

What is a line of fit?

Near the center of the data points

Where should a good line of fit pass?

Estimate values based on the data pattern

What can you use a line of fit to do?

The estimated change in y for each increase of one in x

What does the slope of a line of fit represent?

The predicted y-value when x is zero

What does the y-intercept of a line of fit represent?

A positive slope

What kind of slope does a line of fit usually have for a positive trend?

Fairly evenly above and below the line

How should data points be distributed around a good line of fit?

The observed y-value minus the predicted y-value

What is a residual?

Residual = observed value − predicted value

What is the formula for a residual?

The observed point is above the line of fit

What does a positive residual mean?

The observed point is below the line of fit

What does a negative residual mean?

2

What is the residual if the observed value is 12 and the predicted value is 10?

−2

What is the residual if the observed value is 7 and the predicted value is 9?

y = 1/2x − 4

Rewrite 2x − 4y = 16 in slope-intercept form.

y = 2x + 6

Rewrite y − 4 = 2(x + 1) in slope-intercept form.

y = 1/4x − 5

Rewrite y + 3 = 1/4(x − 8) in slope-intercept form.

y = 5/3x + 4

Rewrite −5x + 3y = 12 in slope-intercept form.

Parallel because both lines have slope 5

Are y = 5x − 2 and y = 5x + 9 parallel perpendicular or neither?

Perpendicular because their slopes are 1/2 and −2

Are y = 1/2x − 2 and 2x + y = 4 parallel perpendicular or neither?

Neither because the slopes are not equal and are not negative reciprocals

Are y = −x − 2 and y = 3x + 3 parallel perpendicular or neither?

Positive correlation

The plotted points rise from left to right. What type of correlation do they show?

Negative correlation

The plotted points fall from left to right. What type of correlation do they show?

No correlation

The plotted points have no clear upward or downward pattern. What type of correlation do they show?

Positive correlation

As the number of hours studied increases test scores tend to increase. What type of correlation is this?

Negative correlation

As the outside temperature increases heating costs tend to decrease. What type of correlation is this?

Strong positive correlation

A scatter plot has points clustered closely around an upward-sloping line. Describe the correlation.

No correlation

A scatter plot has points widely scattered with no clear pattern. Describe the correlation.

An increasing trend between x and y

A data set has points clustered near a line with a positive slope. What does the line of fit show?

A decreasing trend between x and y

A data set has points clustered near a line with a negative slope. What does the line of fit show?

−3

A line of fit predicts y = 10 when x = 4. The observed y-value is 7. What is the residual?

2

Find the slope of the line through (2 3) and (6 11).

−2

Find the slope of the line through (−1 5) and (3 −3).

Undefined

Find the slope of the line through (4 7) and (4 −2).

0

Find the slope of the line through (−3 6) and (5 6).

3

A line rises 9 units while moving 3 units to the right. What is its slope?

−2

A line falls 8 units while moving 4 units to the right. What is its slope?

−3/2

Find the slope of 3x + 2y = 8.

2

Find the slope of the line through (0 −4) and (5 6).

2 units

A line has slope 1/3. How far does it rise when it moves 6 units to the right?

−4

What is the slope of a line parallel to y = −4x + 7?

y − 5 = 3(x − 2)

Write the equation in point-slope form for a line through (2 5) with slope 3.

y − 1 = −2(x + 4)

Write the equation in point-slope form for a line through (−4 1) with slope −2.

y + 3 = 1/2(x − 6)

Write the equation in point-slope form for a line through (6 −3) with slope 1/2.

Slope is 2 and the equation is y − 2 = 2(x − 1)

Find the slope then write the point-slope equation for the line through (1 2) and (4 8).

Slope is −2 and the equation is y − 5 = −2(x + 2)

Find the slope then write the point-slope equation for the line through (−2 5) and (2 −3).

y − 7 = 4(x − 0)

Write the point-slope equation for a line through (0 7) with slope 4.

y − 0 = −1/3(x − 3)

Write the point-slope equation for a line through (3 0) with slope −1/3.

y − 4 = 0(x − 5)

A line passes through (5 4) and has slope 0. Write its equation in point-slope form.

x = −1

A line passes through (−1 −6) and has an undefined slope. What is its equation?

y = 4x + 14

Rewrite y − 2 = 4(x + 3) in slope-intercept form.