Linear Equations and Their GraphsChapter 6

Section 1: Rate of Change and Slope•The dependent variable is the one that

depends on what is plugged in for the other variable (typically y)

•The independent variable is the one that you choose values and plug in (typically x)

•The rate of change is the comparison of the dependent variable and independent variable (change in the dependent variable ÷ change in the independent variable)

•Rate of change usually has units as part of answer

•In a piecewise graph, the rate of change can change, but in a straight line it is constant (called slope)

•Formula for slope:

•We typically leave slope in simplified fraction form (even if it is improper)

•Horizontal lines have a slope of 0•Vertical lines have an undefined slope•Ex1. A balloon is 210 feet above ground

after 5 minutes and 350 feet above ground after 9 minutes. Find the rate of change for the balloon over time.

•Ex2. Find the slope between (-4, 8) and (12, -13)

•Ex3. See “Check Understanding” #2 at the bottom of page 283

2 1

2 1

y yrisem

run x x

Section 2: Slope-Intercept Form•Slope-intercept form is one way to write the

equation of a line•Slope-intercept form is y = mx + b

▫The slope is m▫The y-intercept (where the line crosses the y-

axis is b)• To graph in this form, place the y-intercept

on the graph and then move from that point as the slope dictates (rise over run)• Rule for my classes: No more than 2 graphs

on any coordinate grid

•You must label the axes and the value of the lines

•Ex1. Write an equation of a line with the given slope and y-intercept. m = -3 and b = 7

•Ex2. Find the slope and y-intercept of•If the slope is negative, either the

numerator OR the denominator is negative, not both▫It does NOT matter which one

26

9y x

•Graph each of the following lines•Ex3. Ex4. y = -x – 3 3

75

y x

Section 3: Standard Form•Another form for an equation of a line is

standard form•Standard form: Ax + By = C

▫A, B and C are integers and A cannot be negative

• When graphing in standard form you MUST make a table of at least 3 values• Find the x-intercept, the y-intercept and one

other “test point”• To find the x-intercept, plug in 0 for y and

solve• To find the y-intercept, plug in 0 for x and

solve• You choose the 3rd point to test your line

•Make a table of values and graph each line•Ex1. 8x – 4y = 24 Ex2. 12x + 8y = -

72

•Vertical lines have no y in the equation: x = h•Horizontal lines have no x in the equation: y

= k•Write each equation in standard form.•Ex3.

•Ex4.

38

5y x

26

3y x

Section 4: Point-Slope Form and Writing Linear Equations•A third form for an equation of a line is point-

slope form•Point-slope form:

▫The slope is m and coordinates of the given point come from x1 and y1

• To graph in point-slope form, determine the coordinates of the point and place them on the graph. Then use the slope to find other points on the line

•Ex1. Give the equation of the line in point-slope form that passes through (-3, 7) and slope of 5

1 1( )y y m x x

•Ex4. A line passes through (-3, 7) and (5, 4). Write an equation for the line in point-slope form. Then rewrite the equation in slope-intercept form.

•Open your book to page 304, we are going to do the lesson preview #1-3 together

•Ex5. Write one equation of the line through (3, 8) and (-1, 1) in point-slope form and one in standard form.

•Graph each equation•Ex2. Ex3. 3

5 ( 6)4

y x 2

3 ( 1)5

y x

Section 5: Parallel and Perpendicular Lines

•Two lines are parallel if they lie in the same plane and never intersect

•Parallel lines have the same slope•Ex1. Are the graphs of 4x – 3y = 12 and

parallel? Explain.•Ex2. Write an equation for the line that

contains (7, -2) and is parallel to y = 2x – 8 •Perpendicular lines intersect at 90° angles•The slopes of perpendicular lines are

negative reciprocals (e.g. 2 and -.5): they multiply to be -1

45

3y x

•Ex3. Are the graphs of 3x – 4y = 12 and perpendicular? Explain.

•Ex4. Find an equation of the line that contains (6, 7) and is perpendicular to y = 3x – 5

•Symbol for parallel lines: || or ⁄ ⁄•Symbol for perpendicular lines:

48

3y x

Section 6: Scatter Plots and Equations of Lines•A scatter plot is a graph of discrete points of

data•A trend line is a line drawn through the best

estimate of the center of the data▫It shows the relationship between two sets of

data▫Trend lines are also called linear regression

models • The mathematically best trend line is called

the line of best fit (graphing calculators can calculate this)• Graphing calculators can also give you the

correlation coefficient: a number that tells you how close the data is to falling in a straight line

•The correlation coefficient r is a number between -1 and 1, inclusive

•The closer the absolute value is to 1, the stronger the correlation (the closer it is to a straight line)

•A positive correlation indicates a positive slope

•A negative correlation indicates a negative slope

•To use the graphing calculator:▫Press STAT and then choose EDIT▫Place the x-values in L1 and the y-values in L2▫Press 2nd and then QUIT to go the main screen▫Press STAT again, arrow over to CALC and

choose LINREG▫Press ENTER to have it calculate (r is there

too)

•You must use a graphing calculator to find the line of best fit, but if you are just asked to find a trend line, point-slope form will be the easiest

•Ex1. See “Check Understanding” at the top of page 319

•Ex2. See “Check Understanding” at the top of page 320 (you will need a graphing calculator to do this question)

Section 7: Graphing Absolute Value Equations

•We have graphed absolute value equations before (the graphs are v-shaped)

•We are going to translate the original absolute value function easily (without making a table)

•If there is a number being added or subtracted outside of the absolute value of x, that is the number of places to translate the graph up or down (+ is up and – is down)

•The graph of y = |x|+4 is the v-shaped graph translated 4 places up

•If there is a number being added or subtracted inside of the absolute value of x, that is the number of places to translate left or right (+ is left and – is right)

•The graph of y = |x – 5| is the v-shaped graph translated 5 places right

•Describe how each graph is translated from the parent function y = |x|

•Ex1. y = |x| – 3 •Ex2. y = |x – 7|

•Graph each function by translating y = |x|•Ex3. y = |x + 5| Ex4. y = |x| + 3