Equations of Linear Functions

Chapter 4

Graph Linear Equations(we have already done)

Two points determine a line

Standard Form

Ax + By = CFind the x and

the y-intercepts

An equation represents an infinite number of points in a relationships When given an

equation, make a T-chart substitute the

domain (x values) and find the corresponding range (y-values)

Graph with Slope –Intercept Form

A point and a slope can name a line

y = mx + bplot the y-interceptuse the slope to find more points

Examples

Y = 3/4 x – 23x + 2y = 6-2x + 5y = 102y = 1

Types of Linear Graphs

IncreasingDecreasin

gZero Slope

Undefined Slope or NO slope

y = (positive number)x + b

y = (negative number)x + b

y = constant (domain is all real numbers and the range is the constant)

X = constant (a vertical line is not a function so there is no y-intercept form for it

Families of Linear Graphs y = x (Identity Function)

Y = x + 7 y = x + 5 Y = x – 1 Y = x – ¾Change the slope Y = 1/3xy = 4xY = 10xY = -5x

Change both

Y = 1/3x + 7Y = -3/4x -5Y = 8x -2Y = -4x – 3Y = 5/6x + 9

Change the intercept

Writing Equations in y=mx+bWHEN GIVEN A POINT AND A SLOPE (NOT THE Y-INTERCEPT)

Given: Pt (2,1) and

slope 3 Pt((4, -7) and

slope -1 Pt ((2,-3) and

slope 1/2

WHEN GIVEN TWO POINTS

Given: (3,1), (2,4) (-1, 12), (4, -8)

(5,-8), (-7, 0)

Find an equation for the line.

Given point (3,-2) and slope ¼

Given point (-2, 1) and slope -6

y + 2 = ¼(x – 3) y –(-2)= ¼(x – 3)

y – 1 = -6(x + 2) y – 1 = -6(x –(-2))

Point Slope Form

y – y1 = m(x – x1), where (x1, y1) is a specific point

Where does this equation come from?

m = y1 – y2

x1 – x2

Forms of Linear Equations

Standard FormSlope-

InterceptPoint-Slope

Ax +By = Cy = mx + by – y1= m(x-x1)

Given pt (-5, 3) and m= -2/3

Find the equation in: Point slope form Standard form Slope-intercept form

Which Form to USE???

You need to know how to identify key elements from each type of equation and when to use each!

Group these equations according to similarities and differences

y = 2x – 4y = -3/4x + 3

y = ½ x – 7y = -1/2 x + 2

y = -2x + 5

y = -3/4 xy = -3x + 4y = 4/3 x – 1y = 2x + 5y = .5x - 3

Parallel Lines

Parallel lines have the same slope

Write an equation for a line that passes through the point (-3, 5) parallel to the line y = 2x - 4

Write and equation for a line passing through the point (4,-1) and parallel to the line y = ¼ x + 7

Intersecting Lines

Intersecting lines have different slopes

Write an equation for a line that intersects the line y = -2/3 x + 5 and goes point (-1, 3)

Write an equation for a line that intersects the line 3x – 4y = 10

Perpendicular Lines

The slopes of perpendicular lines are opposite reciprocals

Write and equation for a line that passes through the point (-4,6) and is perpendicular to the line 2x + 3y = 12

Write an equation to a line that passes through the point (4,7) and is perpendicular to the line y = 2/3 x - 1

Scatter Plot

Bivariate DataRegression Lines (line of best fit)CorrelationCausationCorrelation coefficient (r factor)

Inverses~ Where have you seen them?

Additive Inverse (opposite)Multiplicative Inverse (reciprocal)

Square Root (undoes squaring)

Solving Equations

Inverse Relations

If one relation contains the element (a,b), then the inverse relation will contain the element (b,a)

EX: A B(-3, -6) (-6, -3)(-1, 4) (4, -1)(2, 9) (9, 2)((5, -2) (-2, 5)~Display as a set of ordered pairs, Table, Mapping, Graph

Graph the Inverse Functions

“Mathalicious example”~ wins per million we reversed to millions per win

y= x + 3 y =2x + 3 y = -1/3x + 2 y = -3/4x -1

Finding Inverse Functions To find the inverse function f-1 (x) of

the linear function f(x), complete the following steps: Step 1~ Replace f(x) with y in the equation f(x)

Step 2~ Interchange y and x in the equation

Step 3~ Solve the equation for y

Step 4~ Replace y with f-1 (x) in the new equation

Find the Inverse Linear Functions

f(x) = 4x – 6

f(x) = -1/2x + 11

f(x) = -3x + 9

f(x) = 5/4x – 3

f-1(x) = x + 6 4f-1(x) = -2x +22

f-1(x) = -1/3x +3

f-1(x) = 4/5 x + 12/5

Real World Inverse Function Mathalicious example”~ wins per million

we reversed to millions to win f(x)= .103x – 2.96 (NFL cost verses wins) F-1(x) = 9.7x + 2.87 (NFL wins verses

cost) Celsius verse Fahrenheit

C(x) = 5/9(x – 32 C-1(x) = F(x) (Fahrenheit)

Car rental cost per day C(x) = 19.99 + .3x C-1 (x) = total number of miles