Understanding the Turning Point of the Graph for f(x) -x^2 4x - 7

Introduction

The turning point of a graph, often referred to as the vertex of a parabola, is a critical aspect of understanding quadratic functions. This article will delve into the specifics of determining the turning point for the function f(x) -x^2 4x - 7, which is a quadratic equation.

Deriving the Turning Point

The first step in finding the turning point is by differentiating the given function f(x) -x^2 4x - 7:

f'(x) -2x 4

To find the x-coordinate of the turning point, we set the derivative equal to zero:

-2x 4 0

Solving for x yields:

2x 4 x 2

With the x-coordinate of the vertex known, we substitute this back into the original function to find the y-coordinate:

f(2) -(2)^2 4(2) - 7 -4 8 - 7 -3

Therefore, the turning point of the graph is at (2, -3).

Vertex Form of a Parabola

The turning point of a quadratic function can also be found using the vertex form of a parabola:

f(x) -x^2 4x - 7 -x^2 4x - 7 -x^2 - 4x 4 - 4 - 7 -x^2 - 4x 4 - 11 - (x^2 4x) 4 - 11 - (x 2)^2 4 - 11 - (x 2)^2 - 3

The vertex form of a parabola is f(x) a(x - h)^2 k, where (h, k) is the vertex. Thus, the turning point is at (-2, -3)."

Discriminant Method

Another approach to finding the turning point involves the discriminant, which can help determine the maximum or minimum value of a quadratic function. For the given function f(x) -x^2 4x - 7, we first find the discriminant:

D b^2 - 4ac (4)^2 - 4(-1)(-7) 16 - 28 -12

The x-coordinate of the turning point can be found using the formula:

x -b / (2a) -4 / (2(-1)) 2

The y-coordinate of the turning point can be found by substituting x 2 into the original equation, as shown in the previous method.

Conclusion

The turning point of the graph for f(x) -x^2 4x - 7 is at (2, -3). This point represents the maximum value of the function because the parabola opens downwards (since the coefficient of x^2 is negative).