The Vertex of a Parabola
Given a function of the form:
[tex]f(x)=ax^2+bx+c[/tex]Its graph has a shape known as a parabola. The vertex of a parabola is the point of its maximum or minimum value.
The x-coordinate of the vertex is given by:
[tex]x_v=-\frac{b}{2a}[/tex]Given the function:
[tex]f(x)=x^2+bx+c[/tex]It's evident that a =1, but we don't have b or c.
Calculating xv:
[tex]x_v=-\frac{b}{2}[/tex]We are given this value is -5, thus:
[tex]\begin{gathered} -\frac{b}{2}=-5 \\ \text{Solving for b:} \\ b=10 \end{gathered}[/tex]Substitute the value of b in the function:
[tex]y=x^2+10x+c[/tex]We are also given the value of y = 6 when x=-5. Substituting:
[tex]6=(-5)^2+10(-5)+c[/tex]Operating:
[tex]\begin{gathered} 6=25-50+c \\ 6=-25+c \\ c=31 \end{gathered}[/tex]The required coefficients are b = 10 and c = 31
