WRITE A LINEAR FUNCTION THAT SATISFIES THE GIVEN CONDITIONS

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

Problem 1 :

f(-5) = -1 and f(2) = 4

Solution :

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)

Slope (m) = (y₂-y₁) / (x₂-x₁)

m = (4+1) / (2+5)

m = 5/7

Equation of linear function :

y = mx + b ----(1)

y = (5/7)x + b

The linear function passing through the point (-5, -1), we get

-1 = (5/7)(-5) + b

-1 = -25/7 + b

b = -1 + 25/7

b = (25 - 7)/7

b = 18/7

Applying the value of b in (1), we get

y = (5/7)x + (18/7)

Problem 2 :

f(-3) = 5 and f(6) = -2

Solution :

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

When x = -3, y = 5 ==> (-3, 5)

When x = 6, y = -2 ==> (6, -2)

Slope(m) = (y₂-y₁) / (x₂-x₁)

m = (-2-5) / (6+3)

m = -7/9

Equation of linear function :

y = mx + b ----(1)

y = (-7/9)x + b

The linear function passing through the point (-3, 5), we get

5 = (-7/9)(-3) + b

5 = 7/3 + b

b = 5 - 7/3

b = (15 - 7)/3

b = 8/3

Applying the value of b in (1), we get

y = (-7/9)x + (8/3)

Problem 3 :

f(-4) = 6 and f(-1) = 2

Solution :

Let y = mx + bbe the required linear function

y = mx + b --- (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)

Slope (m) = (y₂-y₁) / (x₂-x₁)

m = (2-6) / (-1+4)

m = -4/3

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

y = (-4x/3) + b

Applying (-4, 6), we get

6 = -4(-4)/3 + b

6 = (16/3) + b

b = 6 - 16/3

b = 2/3

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

y = (-4x/3) + (2/3)

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

Problem 4 :

f(1) = 2 and f(5) = 7

Solution :

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

When x = 1, y = 2 ==> (1, 2)

When x = 5, y = 7 ==> (5, 7)

Slope (m) = (y₂-y₁) / (x₂-x₁)

m = (7-2) / (5-1)

m = 5/4

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

y = (5x/4) + b

Applying (1, 2), we get

2 = 5/4 + b

b = 2 - (5/4)

b = 3/4

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

y = (5x/4) + (3/4)

So, the required linear function is y = (5x/4) + (3/4).

Problem 5 :

f(0) = 3 and f(3) = 0

Solution :

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

When x = 0, y = 3 ==> (0, 3)

When x = 3, y = 0 ==> (3, 0)

Slope(m) = (y₂-y₁) / (x₂-x₁)

m = (0-3) / (3-0)

m = -3/3

m = -1

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

y = -x + b

Applying (0, 3), we get

3 = 0 + b

b = 3

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

y = -x + 3

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

Problem 6 :

f(-4) = 0 and f(0) = 2

Solution :

When x = -4, y = 0 ==> (-4, 0)

When x = 0, y = 2 ==> (0, 2)

b = (y₂-y₁) / (x₂-x₁)

b = (2-0) / (0+4)

b = 2/4

b = 1/2

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

y = (x/2) + b

Applying (-4, 0), we get

0 = -4/2 + b

0 = -2 + b

b = 2

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

y = (x/2) + 2

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