Question

What is the y-value of the vertex of the function f(x) = –(x + 8)(x – 14)?

Answer 1

`\bf f(x)=-(x+8)(x-14)\implies f(x)=-(x^2-6x-112) \\\\\\ f(x)=-x^2+6x+112\\\\ -------------------------------\\\\ \textit{vertex of a parabola}\\ \quad \\ \begin{array}{lccclll} f(x)=&-1x^2&+6x&+112\\ &\uparrow &\uparrow &\uparrow \\ &a&b&c \end{array}\qquad \left(-\cfrac{{{ b}}}{2{{ a}}}\quad ,\quad {{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}\right)`

so the y-coordinate is then
`\bf {{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}`

Answer 2

y-value of the vertex of the given function is 121.

Explanation:

Given function f(x) = -( x + 8 )( x - 14 )

We need to find y-coordinate of the vertex of the function.

let, y = -( x + 8 )( x - 14 )

y = -( x( x - 14 ) + 8( x - 14 ) )

y = -( x² - 14x + 8x - 112 )

y = -( x² - 6x - 112 )

Clearly this equation is of parabola. So, now we use completing the square method to write this equation in standard equation of parabola.

y = -( x² - 6x + 3² - 3² - 112 )

y = -( x - 3 )² - ( -9 - 112 )

y = -( x - 3 )² + 121

y - 121 = -( x - 3 )²

( x - 3 )² = -( y - 121 )

y-coordinate of the vertex of the parabola is 121.

Therefore, y-value of the vertex of the given function is 121.