An umbrella sprinkler is positioned on a ceiling at a point whose x coordinate is 0 . Negative values of x indicate distances, in meters, to the left of the position of the sprinkler, and positive values indicate distances to the right. The path of water from the sprinkler can be modeled by the quadratic function w(x)=-(1)/(4)(x^(2)-12) where w(x) is the height of the water, in meters, at position x. Which of the following equivalent expressions displays the height of the ceiling as a constant or coefficient? Choose 1 answer:

Question

An umbrella sprinkler is positioned on a ceiling at a point whose  x coordinate is 0 . Negative values of  x indicate distances, in meters, to the left of the position of the sprinkler, and positive values indicate distances to the right. The path of water from the sprinkler can be modeled by the quadratic function  w(x)=-(1)/(4)(x^(2)-12) where  w(x) is the height of the water, in meters, at position  x. Which of the following equivalent expressions displays the height of the ceiling as a constant or coefficient? Choose 1 answer:

Bytelearn · Accepted Answer

Analyze Function w(x): Analyze the given quadratic function w(x) = -(1/4)(x^2 - 12). This function represents the height of the water at position x. To find the height of the ceiling, we need to find the maximum value of w(x), which occurs at the vertex of the parabola represented by the quadratic function.Find Vertex Form: The quadratic function is in the form w(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. To find the equivalent expression that displays the height of the ceiling as a constant or coefficient, we need to complete the square to rewrite the given function in vertex form.Factor Out Coefficient: The given function is w(x) = -(1/4)(x^2 - 12). To complete the square, we need to factor out the coefficient of x^2 from the x-terms. Since the coefficient is already factored out, we can proceed to complete the square.Complete the Square: The term that completes the square is derived from the formula (b/2a)^2, where a is the coefficient of x^2 and b is the coefficient of x. However, since there is no x term in the given function, we do not need to add or subtract any term to complete the square. The function is already in a form where the constant term -12 can be seen as the term that has been added to complete the square.Equivalent Expression: The equivalent expression in vertex form is w(x) = -(1/4)(x^2) + 3, where 3 is the constant term that represents the height of the ceiling. This is because the vertex form of the parabola is w(x) = a(x - h)^2 + k, and the maximum value of w(x) is k when a is negative, which is the case here.

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