CA 6th ed HLR1.4 F15 O’Brien
1.4: Equations of Lines and Linear Models
I.Summary of Key Concepts about Lines
A.Slope:
B.Horizontal Lines
1.Equation: f(x) = b.
2.m = 0
3.No x-intercept
4.A horizontal line is a constant (0 degree) function.
5.The domain of a horizontal line is .
6.The range of a horizontal line is .
C.Vertical Lines
1.Equation: x = a
2.m is undefined
3.No y-intercept
4.A vertical line is not a function.
5.The domain of a vertical line is {a}.
6.The range of a vertical line is .
D.Oblique Lines
1.Oblique lines are true linear (1st degree) functions. They cut the x axis and the y axis.
2.Slope-intercept Form: f(x) = mx + b or y = mx + b
a.“m” = slope
b.“b” = y-intercept
3.Standard Form: Ax + By = C or
a.slope =
b.y-intercept =
4.Point-Slope Form:
a.slope = m
b.(x1, y1) = a point on the line
E.Parallel Lines
1.Parallel lines have the same slope.
2.If the slope of a line is , the slope of a line parallel to it is also .
F.Perpendicular Lines
1.Perpendicular lines have negative reciprocal slopes.
2.If the slope of a line is , the slope of a perpendicular line is .
3.The product of the slopes of two perpendicular line is –1.
II.Writing the Equation of a Line
To write the equation of a line, we need to know three things:
1.a point: (x1, y1)
2.the slope: m
3.a formula: either the slope-intercept form y = mx + b or the point-slope form
y – y1 = m(x – x1)
A.How to Write the Equation of a Line Given a Point and a Slope Example 1
1.Method One: Using the Point-Slope Form (PSF)
a.Plug the given point, (x1, y1), and the given slope, m, into the Point Slope Form
y – y1 = m(x – x1).
b.Solve the resulting equation for y.
2.Method Two: Using the Slope-Intercept Form (SIF)
a.Plug the given point (x, y) and the given slope, m, into the Slope-Intercept Form
y = mx + b.
b.Solve the resulting equation for b.
c.Write the requested equation by plugging the actual values of m and b into
the formula y = mx + b.
Caution: Do not forget step c. You aren’t done with the problem until you have
written a linear equation.
3.You do not have to do both methods. Choose your favorite and stick with it.
B.How to Write the Equation of a Line Given Two Points Examples 2 & 3
1.Find the slope by plugging the two given points into the slope formula.
2.Use method one or method two to write the equation of the line.
C.How to Write the Equation of a Line Given a Point and the Equation of a Parallel Line Example 4
1.Solve the given equation for y and identify the slope of the given line.
2.Use the given point and the slope you found in step 1 to write the equation of the line.
You may use either method.
D.How to Write the Equation of a Line Given a Point & the Equation of a Perpendicular Line Example 5
1.Solve the given equation for y and identify the slope of the given line.
2.Find the negative reciprocal of the slope of the given line.
3.Use the given point and the slope you found in step 2 (the negative reciprocal) to write
the equation of the line. You may use either method.
E.Application Problems Example 6
1.From the given information, identify either two points or a point and a slope.
2.Use the 2 points or point & slope you found in step 1 and either method to write the
linear model.
III.Regression Analysis on a TI-82, 83, or 84 Graphing Calculator
1.Clear Old Data [Omit Step 1 if there is no old data in L1 and/or L2.]
STAT 1:Edit ▲ (highlight L1) Clear Enter ► ▲ (highlight L2) Clear Enter ◄
2.Enter Data
STAT 1:Edit (Enter input (x) values in L1 and output (y) values in L2.) Make sure every entry is correct.
Example 7
Average annual tuition and fees for in-state students at public 4-year colleges for selected years are shown in the
table. Using x = 0 for 1990, enter the x-values (years since 1990) in L1 and the y-values (tuition and fees) in L2.
Year / Years since 1990 (x)L1 / Tuition & Fees (y)
L2
1990 / 0 / 2035
1994 / 4 / 2820
1996 / 6 / 3151
1998 / 8 / 3486
2000 / 10 / 3774
3.Create a Scatter Plot of the Data
y = CLEAR (Clear out or turn off any equations.)
2nd y = 1:Plot 1 (to select Plot 1) Enter (to turn Plot 1 on)
ZOOM 9:ZoomStat
4.Run Regression
STAT ► CALC
select appropriate regression LinReg (ax + b) QuadReg CubicReg QuartReg etc.
Enter (to execute regression analysis)
LinReg
y = ax + bRecord the model as an equation (you may round to three decimal places).
a = 174.3986486y = 174.399x + 2076.567
b = 2076.567568If given, record the correlation coefficient, r (do not round).
r2 = .9961354296r = .9980658443 (We will not be using r2, the correlation of determination.)
r = .9980658443Tell the goodness of fit (The closer r is to 1 or –1, the better the fit).
Very good fit
TI-83 and TI-84 users
If you do not see r and r2, do the following: Press 2nd 0 (catalog) and scroll down until you see DiagnosticOn.
Hit enter to select that command and then hit enter again to execute it. Run the regression again. You should
now see r and r2. Once you’ve turned the diagnostic on, it should stay on. So you only have to do this
procedure once.
5.Graph the Model with the Data
y = VARS 5:Statistics ►► RegEQ GRAPH
Create an accurate sketch of the model with the data.
If the type of regression you ran does not give a correlation coefficient, you can assess
the goodness of fit by examining the graph of the model with the data. The more data
points the model goes through or touches, the better the fit.
6.To Find the Output for a Specified Input
2nd Graph (Table) Enter appropriate input.
For example, by plugging in 18 we can get the projected cost of tuition and fees in 2008.
y(18) = $5215.70
7.Turn the Plotter Off
2nd y = 4 Enter (It should say PlotsOff Done.)
1