Given The Function F(x) = A(x-h)^2 + K Where A And K Are Negative, Which Of The Following Cannot Be True: A) F(5) < 0, B) F(-5) < 0, C) F(1) = K, D) F(0) = -k?

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In mathematics, quadratic functions play a crucial role in various applications, from modeling projectile motion to optimizing business processes. One standard form of a quadratic function is given by f(x) = a(x-h)^2 + k, where a, h, and k are constants that determine the shape and position of the parabola. This article delves into the properties of this function, particularly focusing on the scenario where a and k are negative. We will analyze the implications of these conditions and determine which statements about the function's behavior cannot be true.

The Anatomy of f(x) = a(x-h)^2 + k

The function f(x) = a(x-h)^2 + k is a quadratic function expressed in vertex form. Each constant in the equation provides valuable information about the parabola's characteristics:

  • a: The coefficient a determines the direction and width of the parabola. If a > 0, the parabola opens upwards, and if a < 0, the parabola opens downwards. The magnitude of a also affects the parabola's width; a larger absolute value of a results in a narrower parabola, while a smaller absolute value leads to a wider parabola. In our case, we are given that a is negative, indicating that the parabola opens downwards, resembling an inverted U-shape.
  • h: The constant h represents the x-coordinate of the vertex of the parabola. The vertex is the point where the parabola changes direction – either the minimum point if the parabola opens upwards or the maximum point if the parabola opens downwards. The vertex form of the quadratic equation makes it easy to identify the vertex, which is located at the point (h, k). The value of h dictates the horizontal shift of the parabola from the basic parabola y = ax^2. A positive h shifts the parabola to the right, and a negative h shifts it to the left.
  • k: The constant k represents the y-coordinate of the vertex of the parabola. It indicates the vertical shift of the parabola from the basic parabola y = a(x-h)^2. A positive k shifts the parabola upwards, and a negative k shifts it downwards. In the context of our problem, k is negative, which means the vertex of the parabola lies below the x-axis.

Implications of Negative a and k

Given that a and k are both negative, we can deduce several key properties of the function f(x) = a(x-h)^2 + k. The negative value of a tells us that the parabola opens downwards. This means that the vertex of the parabola represents the maximum point of the function. Since k is also negative, the y-coordinate of the vertex is negative, placing the vertex below the x-axis. The vertex is the point (h, k), and since k < 0, the highest point of the parabola is below the x-axis.

Because the parabola opens downwards and its vertex is below the x-axis, the function values will be negative for a wide range of x-values. The function will only be positive (if at all) in a limited interval around the axis of symmetry, which is the vertical line x = h. If the vertex is sufficiently far below the x-axis, the function may never be positive. If the parabola intersects the x-axis, the function will have two real roots; otherwise, it will have no real roots.

Analyzing the Given Statements

Now, let's consider the given statements and determine which one cannot be true when a and k are negative:

A) f(5) < 0: This statement asserts that the function value at x = 5 is negative. Since the parabola opens downwards and the vertex is below the x-axis, it is plausible for f(5) to be negative. Depending on the specific values of a, h, and k, the point (5, f(5)) could indeed lie below the x-axis. For instance, if the vertex is far to the left or right of x = 5, and the parabola is narrow enough, f(5) would be negative.

B) f(-5) < 0: Similar to statement A, this statement claims that the function value at x = -5 is negative. Again, this is a possible scenario. If the parabola's vertex is located such that x = -5 is far from the vertex, the function value at x = -5 could be negative. The position of the vertex and the width of the parabola, determined by a, will influence the function value at x = -5.

C) f(1) = k: This statement is particularly interesting. It suggests that the function value at x = 1 is equal to k, the y-coordinate of the vertex. Let's analyze this using the function's equation:

f(1) = a(1-h)^2 + k

For f(1) to equal k, the term a(1-h)^2 must be equal to zero:

a(1-h)^2 = 0

Since a is negative and therefore not zero, the only way for this equation to hold true is if (1-h)^2 = 0. This implies that 1 - h = 0, which means h = 1. Therefore, f(1) = k is true only if the x-coordinate of the vertex, h, is equal to 1. This is a specific condition, but it is possible.

D) f(0) = -k: This statement asserts that the function value at x = 0 is equal to -k. Given that k is negative, -k would be a positive value. Let's examine the implications:

f(0) = a(0-h)^2 + k = ah^2 + k

For f(0) to equal -k, we must have:

ah^2 + k = -k

ah^2 = -2k

Since a is negative and k is negative, -2k is positive. However, ah^2 is the product of a negative number (a) and a non-negative number (h^2), making ah^2 a negative value or zero. Therefore, ah^2 cannot be equal to -2k, which is a positive value. This condition cannot be met.

Conclusion

After analyzing the given statements, we can conclude that statement D, f(0) = -k, cannot be true when a and k are negative. The other statements are possible under certain conditions, but f(0) = -k leads to a contradiction given the constraints on a and k. Understanding the properties of quadratic functions, particularly the roles of the constants a, h, and k, is essential for solving problems involving these functions.

Therefore, the answer is D) f(0) = -k.