Question

The function r(x) = x^2 + 6x + 9. What is the vertex of g(x) = r(x) - 2?

Answer 1

Answer:

The vertex of g(x) is (-3, -2).

Step-by-step explanation:

To find what g(x) truly represents, we must substitute r(x) with the function itself to get the derived function g(x).

g(x) = r(x) - 2

g(x) = x² + 6x + 9 - 2    (Simplify)

g(x) = x² + 6x + 7    (Now, we can start to find the vertex of this quadratic function)

The vertex can be found, when the function is in standard form (AX² + BX + C), by this defined ordered pair: $(\frac{-b}{2a}, y)$

Notice that we already know what A, B, and C are, so we don't need to worry about them, but we don't have the y value. This can be calculated by plugging in the x value for the axis of symmetry which is (-b) / (2a). The vertex is always at the axis of symmetry and so, by plugging in the x value for the axis of symmetry, we can find the exact y value for the location of the vertex on the cartesian plane.

A = 1, B = 6, C = 7 in the quadratic function g(x).

The x value for the vertex can be found by:

x = (-6) / 2(1)    (Simplify)

x = -3

Now that we have x, we can find y by substituting this into g(x):

g(x) = (-3)² + 6(-3) + 7    (Simplify)

g(x) = 9 - 18 + 7    (Simplify)

g(x) = -2

The y value for our vertex is -2.

The vertex of g(x) is (-3, -2).