Answer:
C. [tex]f(x) = (x-2.5)^{2}-3.25[/tex]
Step-by-step explanation:
The vertex form for a quadratic function is:
[tex]f(x) = a(x-h)^2+k[/tex]
- Then, you expand the binomial and get:
[tex]f(x) = a(x^2-2hx +h^2) + k[/tex]
- Now you distribute and get the first form:
[tex]f(x) = ax^2-2ahx +ah^2 + k[/tex]
- You know that the general form for a quadratic function is:
[tex]f(x) = ax^2+bx+c, Second\ form[/tex]
- You compare the two forms that you have and finding h and k:
[tex]ax^2+2ahx +ah^2 + k = ax^2+bx+c[/tex]
- Finding h from the coefficient of X:
[tex]-2ah = b[/tex]
[tex]h = -\frac{b}{2a}[/tex]
from the quadratic function given you know that a = 1 , b = -5 and c = 3, thus:
[tex]h = -\frac{(-5)}{2(1)}=2.5[/tex]
- Finding k from the third coefficient:
[tex]ah^2+k = c[/tex]
[tex]Isolate\ k =>\ k = c-ah^2[/tex]
You know c,a and h, so replace the values:
[tex]k = 3-(1)(2.5)^2 \\ k = 3-6.25\\ k = -3.25\\[/tex]
• Finally replace the values for a, h and k in the vertex form:
[tex]f(x) = a(x-h)^2+k\\ f(x) = (1)(x-2.5)^2+(-3.25)\\ f(x) = (x-2.5)^2-3.25[/tex]
So answer is C. [tex]f(x) = (x-2.5)^2-3.25[/tex]
