Question

Write in standard form the quadratic function defined by the points (0, 5), (5, 0), and (3, −4).

Answer 1

`f(x)=x^2-6x+5`

Explanation:

The general form of quadratic function is

`f(x)=ax^2+bx+c` .... (1)

It is given that the function defined by the points (0, 5), (5, 0), and (3, −4). It means the function must be satisfied by these points.

For (0,5),

`5=a(0)^2+b(0)+c\Rightarrow c=5`

The value of c is 5.

For (5,0),

`0=a(5)^2+b(5)+(5)\Rightarrow 25a+5b=-5` .... (2)

For (3,-4),

`-4=a(3)^2+b(3)+(5)\Rightarrow 9a+3b=-9` .... (3)

Multiply equation (2) by 3 and equation (3) by 5.

`75a+15b=-15` .... (4)

`45a+15b=-45` .... (5)

Subtract equation (5) from equation (4).

`75a-45a=-15-(-45)`

`30a=30`

`a=1`

Substitute a=1 in equation (3).

`9(1)+3b=-9`

`9+3b=-9`

`3b=-9-9`

`3b=-18`

`b=-6`

Substitute a=1,b=-6 and c=5 in equation (1).

`f(x)=1x^2+(-6)x+(5)`

`f(x)=x^2-6x+5`

Therefore, the required function is
`f(x)=x^2-6x+5`.