Bell Ringer

1) If (x) = 3x + 2, then what is the solution of f(2). Hint: substitute 2 in for x.

2) If f(x) = 2x2 – 3x + 4, then what is f(3), or what’s the solution when you substitute 3 in for x?

Linear FunctionsLinear Functions

Function Notation-Function Notation-a linear function written in the form a linear function written in the form y = mx + by = mx + b where y is written as a function where y is written as a function f.f.

f(x) = mx + bf(x) = mx + bslopeslope y-intercepty-intercept

x-coordinatex-coordinate

f(x) is another name for y.f(x) is another name for y.It means “the value of f at x.”It means “the value of f at x.”g(x) or h(x) can also be used to name functions g(x) or h(x) can also be used to name functions

This is read This is read as ‘f of x’as ‘f of x’

Domain and Range DomainDomain = values of ‘x’ for which the function is = values of ‘x’ for which the function is

defined.defined.

Range Range = the values of f(x) where ‘x’ is in the = the values of f(x) where ‘x’ is in the domain of the function domain of the function f. f.

The graph of a function The graph of a function f f is the set of all points (x, is the set of all points (x, f(x)). f(x)).

Graphing a FunctionGraphing a Function To graph a function:To graph a function:

(1) (1) make a table by substituting into the make a table by substituting into the function. function.

(2) (2) plot the points from your table and plot the points from your table and connect the points with a line.connect the points with a line.

(3) (3) identify the domain and range, (if identify the domain and range, (if restricted)restricted)

Graph a Function

32)( xxf

xx -2-2 -1-1 00 11 22

f(x)f(x) -7-7 -5-5 -3-3 -1-1 11

STEPSTEP 11

SOLUTION

Graph the Function Graph the Function f(x) = 2x – 3 f(x) = 2x – 3

STEPSTEP 22

Make a table by choosing a few values for x and then finding values for y.

STEPSTEP 33

Plot the points. Notice the points appear on a line. Connect the points drawing a line through them.

32)( xxf

The domain and range are not restricted therefore, you do not have to identify.

Graph a Function1122Graph the functionGraph the function f(x)f(x)

== – – xx + 4+ 4 with domainwith domain x x ≥≥ 0.0.

Then identify the range of the functionThen identify the range of the function..

STEP 1Make a table.

x 0 2 4 6 8

y 4 3 2 1 0

STEP 2

Connect the points with a ray because the domain is restricted.

Plot the points.

STEP 3Identify the range. From the graph, you can see that all points have a y-coordinate of 4 or less, so the range of the function is y ≤ 4.

42

1)( xxf

Family of FunctionsFamily of Functionsis a group of functions with similar characteristics. For example, functions that have the form f(x) = mx + b f(x) = mx + b constitutes the family of linear functions.constitutes the family of linear functions.

Parent Linear FunctionParent Linear Function

The most basic linear function in the family of all linear functions is called the PARENT LINEAR FUNCTION which is:

f(x) = xf(x) = xxx -5-5 -2-2 00 11 33

f(x)f(x) -5-5 -2-2 00 11 33

f(x) = x

Real-Life FunctionsReal-Life Functions

A cable company charges new customers $40 for installation and $60 per month for its service. The cost to the customer is given by the function f(x) = 60x +40 where x is the number of months of service. To attract new customers, the cable company reduces the installation fee to $5. A function for the cost with the reduced installation fee is g(x) = 60x + 5. Graph both functions. How is the graph of g related to the graph of f ?

Real-Life FunctionsReal-Life Functions

The graphs of both functions are shown. Both functions have a slope of 60, so they are parallel. The y-intercept of the graph of g is 35 less than the graph of f. So, the graph of g is a vertical translation of the graph of f.

Write an Equation Given a Slope and a Point

Write the Equation using Point-Slope Write the Equation using Point-Slope FormForm

)6,1(3

1 andm

)( 11 xxmyy Step 1: Plug it in

Point- Slope Form

A ChallengeA Challenge

Write the equation of a line in point-slope form that passes through (-3,6) and (1,-2)

Hint: Find the change in y and the change in x.

Change is determined by subtraction.

Reminder that slope is rise or run.

A ChallengeA Challenge

Can you write the equation of a line in point-slope form that passes through (-3,6) and (1,-2)

8 2 6 + 2 -3 – 1

= - 4

= - 1

Calculate the slope.m =

y – y1 = m(x – x1)

y + 2 = - 2 (x – 1)

Use m = - 2 and the point (1, -2).

y = -2(x – 1) - 2

y = -2x + 2 - 2

y = -2x + 0 y = -2x

Point-Slope Form

Slope-Intercept Form

Equations of Parallel Lines

Write an equation for the line that contains

(5, 1) and is parallel to

m =

45

3 xy

5

3

55

31 xy

Equations of Perpendicular Lines

Find the equation of the line that contains

(0, -2) and is perpendicular to y = 5x + 3

m = 5

1

xy5

12