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

Solution:

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

= -(x² - 112 - 6x)

= -x² + 6x + 112 --- (1)

We know the graph of an equation y = ax² + bx + c where a ≠ 0 is a parabola. The parabola opens upwards if a > 0 and opens downwards if a < 0. The vertex of the parabola is the point where the axis and parabola intersect. Its x coordinate x = -b/2a and its y coordinate is found out by substituting x = -b/2a in the parabola equation.

The parabola given in the problem statement has a negative coefficient of x² and hence it is a parabola which opens downwards. Also for the equation
a = -1, b = 6 and c = 112. Therefor the x-coordinate of the vertex is 

x = -b/2a

= -(6)/[2(-1)]

= 3

Substituting the value of x = 3 in equation (1) we get,

Now y = -(3)² + 6(3) + 112

= -9 + 18 + 112 = 121

So the vertex point coordinates are (3, 121) and the y value is 121.

The graph below verifies the vertex point.

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

Summary:

The y-value of the vertex of the function f(x) = -(x + 8)(x - 14) is 121.