FIND THE LINEAR FUNCTION SATISFYING THE GIVEN CONDITIONS

Example Problems

Example 1 :

Write an equation for the linear function f satisfying the conditions. Graph y = f(x).

(i)  f(-5)  =  -1 and f(2)  =  4

(ii)  f(-4)  =  6 and f(-1)  =  2

Solution :

Let y  =  a+bx be the required linear function

y  =  a+bx ---(1)

By comparing the given question with y  =  f(x), we know that

when x  =  -5, y  =  -1  ==> (-5, -1)

when x  =  2, y  =  4  ==> (2, 4)

b  = (y₂-y₁)/(x₂-x₁)

b  =  (4+1)/(2+5)

b  =  5/7

To find a, we may apply any one of the points and value of b in (1).

y  =  a + (5x/7)

Applying (-5, -1), we get

-1  =  a + 5(-5)/7

-1  =  a - (25/7)

-7+25  =  7a

18  =  7a

a  =  18/7

By applying the value of a and b in (1) we get,

y  =  (18+5x)/7

So, the required linear function is y  =  (5x+18)/7.

(ii)   f(-4)  =  6 and f(-1)  =  2

Solution :

Let y  =  a+bx be the required linear function

y  =  a+bx ---(1)

By comparing the given question with y  =  f(x), we know that

when x  =  -4, y  =  6  ==> (-4, 6)

when x  =  -1, y  =  2  ==> (-1, 2)

b  = (y₂-y₁)/(x₂-x₁)

b  =  (2-6)/(-1+4)

b  =  -4/3

To find a, we may apply any one of the points and value of b in (1).

y  =  a + (-4x/3)

Applying (-4, 6), we get

6  =  a-4(-4)/3

6  =  a+(16/3)

18  =  3a+16

2  =  3a

a  =  2/3

By applying the value of a and b in (1) we get,

y  =  (2-4x)/3

So, the required linear function is y  =  (-4x+2)/3

Example 2 :

 g(2)  =  1 and the graph of g is parallel to the line

6x−3y  =  2

Solution :

Since the required line and the given line are parallel, their slopes will be equal.

g(2)  =  1  ==>  (2, 1)

b  =  -coefficient of x/coefficient of y

b  =  -6/(-3)

b  =  2

By applying the value of b in the general form of linear function, we get 

y  =  a+2x

The required linear function passes through the point (2, 1).

1  =  a+2(2)

1  =  a+4

a  =  -3

y  =  -3+2x

So, the required linear function is y  =  2x-3.

Example 3 :

 g(2)  =  1 and the graph of g is perpendicular to the line

6x−3y  =  2

Solution :

Since the required line and the given line are perpendicular, the product of their slopes will be equal.

g(2)  =  1  ==>  (2, 1)

b  =  -coefficient of x/coefficient of y

b  =  -6/(-3)

b  =  2

Required slope  =  -1/2

By applying the value of b in the general form of linear function, we get 

y  =  a+(-x/2)

The required linear function passes through the point (2, 1).

1  =  a-(2/2)

1  =  a-1

a  =  2

y  =  2-(x/2)

y  =  (4-x)/2

So, the required linear function is y  =  (-x+4)/2.

Example 4 :

The x and y-intercepts of f are 5 and −1, respectively.

Solution :

y  =  a+bx ----(1) 

By writing x intercept as point, we get (5, 0).

By writing y intercept as point, we get (0, -1).

Slope  =  (-1-0)/(0-5)

b  =  1/5

y  =  a+(x/5)

The required function is passing through the point (5, 0).

0  =  a+(5/5)

a  =  -1

y  =  -1+(x/5)

y  =  (-5+x)/5

Example 5 :

A plumber charges a $60 call out fee and $48 per hour.

a) Copy and complete the table :

b)  Name the dependent and independent variables.

c)  Find the linear model that connects the variable.

d)  Find the cost of a plumbing job that took 6  1/4 hours.

Solution :

Given that, a plumber charges a $60 call out fee and $48 per hour.

Let h be the number of hours he is working. Let C be the total cost.

C = 60 + 48h

a)

b)  Number of hours he is working in independent variable and total cost is dependent variable.

c)  The required function is C = 60 + 48h

d)  Cost of a plumbing job that took 6  1/4 hours :

h = 6.25

Applying the value of h, we get

C = 60 + 48(6.25)

= 60 + 300

= 360

For working 6  1/4 hours, he get $360.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.