Answer:
[tex]f(x)=x^2-6x+5[/tex]
Step-by-step explanation:
The general form of quadratic function is
[tex]f(x)=ax^2+bx+c[/tex] .... (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),
[tex]5=a(0)^2+b(0)+c\Rightarrow c=5[/tex]
The value of c is 5.
For (5,0),
[tex]0=a(5)^2+b(5)+(5)\Rightarrow 25a+5b=-5[/tex] .... (2)
For (3,-4),
[tex]-4=a(3)^2+b(3)+(5)\Rightarrow 9a+3b=-9[/tex] .... (3)
Multiply equation (2) by 3 and equation (3) by 5.
[tex]75a+15b=-15[/tex] .... (4)
[tex]45a+15b=-45[/tex] .... (5)
Subtract equation (5) from equation (4).
[tex]75a-45a=-15-(-45)[/tex]
[tex]30a=30[/tex]
[tex]a=1[/tex]
Substitute a=1 in equation (3).
[tex]9(1)+3b=-9[/tex]
[tex]9+3b=-9[/tex]
[tex]3b=-9-9[/tex]
[tex]3b=-18[/tex]
[tex]b=-6[/tex]
Substitute a=1,b=-6 and c=5 in equation (1).
[tex]f(x)=1x^2+(-6)x+(5)[/tex]
[tex]f(x)=x^2-6x+5[/tex]
Therefore, the required function is [tex]f(x)=x^2-6x+5[/tex].
