Explanation
- Formulas for finding the vertex.
[tex]h = - \frac{b}{2a} \\ k = \frac{4ac - {b}^{2} }{4a} [/tex]
[tex]y = a{(x - h)}^{2} + k [/tex]
Where the a-term determines how wide, narrow, upward or downward the graph is.
The h-term determines the horizontal shift of graph. (x-axis)
The k-term determines the vertical shift of graph. (y-axis)
The vertex of graph is at (h,k). We can find the vertex by using the given formulas.
From the equation, the coefficients are:
[tex] \begin{cases} a = - 1 \\ b = 12 \\ c = - 31 \end{cases}[/tex]
Substitute these values in the formulas.
Find the value of h
[tex]h = - \frac{12}{2( - 1)} \\ h = - \frac{12}{ - 2} \\ h = - \frac{ - 6}{1} \longrightarrow - ( - 6) = 6 \\ h = 6[/tex]
Hence, the value of h is 6.
Find the value of k
[tex]k = \frac{4( - 1)( - 31) - {(12)}^{2} }{4( - 1)} \\ k = \frac{124 - 144}{ - 4} \\ k = \frac{ - 20}{ - 4} \longrightarrow \frac{5}{1} = 5 \\ k = 5[/tex]
Hence, the value of k is 5. Then we substitute both value of h and k in the vertex form
- Rewrite the equation of vertex form by substituting the value of h and k
[tex]y = a {(x - h)}^{2} + k \\ y = - 1 {(x - 6)}^{2} + 5 \\ y = - {(x - 6)}^{2} + 5[/tex]
Answer
Since the vertex is at (h,k). Therefore the value of h is 6 and k is 5. Therefore:
[tex] \large{(h, k)=(6, 5)}[/tex]
(6,5) is the vertex.
