Linear Functions

The first degree or linear function f (x) = mx + b or f (x) = ax + b is one in which
both variables, x and y or f (x) , are raised to the first power.

m or a is the slope and b is the y-intercept.

The slope of a linear function is constant.

There are 3 possibilities for the slope of a linear function:

if m = 0 the function is a horizontal line or constant function.

if m is undefined (or ) the line is vertical and not a function.

if m is Real and m is not 0 nor undefined, the line is neither vertical nor horizontal.

if m is positive, the line leans to the right, if m is negative, the line leans to the left.

In depth study of the linear equation/function is covered in lessons 3.1 - 3.4 of the Analytic Geometry MathRoom.

The only difference between the linear function and the linear equation is that

we refer to f (x) instead of y when it's a function.

Now get a pencil, an eraser and a note book, copy the questions,
do the practice exercise(s), then check your work with the solutions.
If you get stuck, review the examples in the lesson, then try again.

Practice

1) Write the letter L beside each linear function:

a) – 5x2 + 3x – 2 = 0	b) y = 7x – 5	c) 4x + 2y – 17 = 0
d) x3 – 7x2 + 3 = 9x	e) 4x = – 9y + 3	f)

2) Write a linear function rule to represent each situation:

a) Harry's age (H) is 2 years less than 3 times Sally's age (s) .

b) The cost (C) of a car rental is $24 per day (d) plus a $35 fixed fee for insurance.

c) The temperature in degrees Celcius (C) is 5/9 of the temperature in Farenheit (F) minus 32°.

d) Sam (S) has 4 times as many baseball cards as John (J ).

e) The base of a rectangle is 4 cm. more than the height (h). Find P(h) the perimeter function.

3) Write a linear function rule to represent the following:

a) the line includes the points (2, –3), (0, 4) and (4, –10).

b) the line has an initial value of 7 and a rate of change (slope) of –2/3.

c)

d) the line with slope = – 8, goes through the y-intercept of y = 3x – 7.

e) the line is perpendicular to y = 4/3 x – 12 and goes through ( 3, –8).

.

Solutions

1) b, c, e are linear functions.

2) Write a linear function rule to represent each situation:

a) Harry's age (H) is 2 years less than 3 times Sally's age (s). H(s) = 3s – 2

b) The cost (C) is $24 per day (d) plus $35 fixed fee. C(d) = 24d + 35

c) The temperature in Celcius is 5/9 of Farenheit minus 32°.

d) Sam (S) has 4 times as many baseball cards as John (J ). S(J) = 4J

e) The base of a rectangle is 4 cm. more than the height (h). Find P(h) the perimeter function.

h = height then h + 4 = base
perimeter = 2(sum of base and height)
P(h) = 2(h + h + 4)
P(h) = 4h + 8

3) Write a linear function rule to represent the following:

a) the line includes the points (2, – 3), (0, 4) and (4, – 10).

slope = , y-intercept = 4 (point (0, 4) is a y-int) so

b) the line has an initial value of 7 and a rate of change (slope) of – 2/3.

c) slope = , using point (–1, 7) we get:

d) the line goes through the y-intercept of y = 3x – 7 and has slope of – 8. f (x) = – 8x – 7

e) the line is perpendicular to y = 4/3 x – 12 and goes through ( 3, – 8).

( Functions MathRoom Index )

(all content of the MathRoom Lessons © Tammy the Tutor; 2004 - ).