The function in vertex form is [tex]f(x)=(x+\frac{1}{2})^{2}+\frac{3}{4}[/tex] ⇒ 3rd answer
Step-by-step explanation:
The vertex form of the quadratic function f(x) = ax² + bx + c is
f(x) = a(x - h)² + k, where
∵ f(x) = x² + x + 1
∴ a = 1 , b = 1
∵ [tex]h=\frac{-b}{2a}[/tex]
- Substitute the values of a and b to find h
∴ [tex]h=\frac{-1}{2(1)}[/tex]
∴ [tex]h=\frac{-1}{2}[/tex]
Substitute the value of x in f(x) by the value of h to find k
∵ f( [tex]\frac{-1}{2}[/tex] ) = [tex](\frac{-1}{2})^{2}+\frac{-1}{2}+1[/tex]
∴ f( [tex]\frac{-1}{2}[/tex] ) = [tex]\frac{1}{4}-\frac{1}{2}+1[/tex]
∴ f( [tex]\frac{-1}{2}[/tex] ) = [tex]\frac{3}{4}[/tex]
- k is the value of f(x) when x = h
∵ h = [tex]\frac{-1}{2}[/tex]
∴ k = f( [tex]\frac{-1}{2}[/tex] )
∴ k = [tex]\frac{3}{4}[/tex]
Substitute the values of a, h and k in the vertex form
∵ f(x) = a(x - h)² + k
∵ a = 1 , [tex]h=\frac{-1}{2}[/tex] , [tex]k=\frac{3}{4}[/tex]
∴ [tex]f(x)=1(x-\frac{-1}{2})^{2}+\frac{3}{4}[/tex]
∴ [tex]f(x)=(x+\frac{1}{2})^{2}+\frac{3}{4}[/tex]
The function in vertex form is [tex]f(x)=(x+\frac{1}{2})^{2}+\frac{3}{4}[/tex]
Learn more:
You can learn more about the quadratic functions in brainly.com/question/9390381
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