Question

Transform the parent quadratic function f(x) = x2 to find a function g(x) that models the given situation. The point C is the vertex of g(x). Restrict the domain of g(x) to values that have meaning in the given situation.A red velvet rope hangs between two stanchions and forms a curve that can be modeled by a parabola. In the illustration shown, the unit of measurement for both axes is feet, and the vertex of the curve is point C. Find a quadratic function of the form g(x) = a f(x − h) + k that models the rope, and state the function's domain.The equation that models the velvet rope is g(x) = .The domain of g(x) is {x|}.

Answer 1

The equation of a parabola is represented as: $y = a(x- h)^2 + k$, where (h,k) represents the vertex.

• The equation of g(x) is $y = 2(x - 3)^2 + 5$
• The domain of the function is: $(-\infty, \infty)$

The question is incomplete, as the vertex and the required coordinates are not given. So, I will make assumptions.

Assume the vertex is (3,-5).

So, we have:

$y = a(x- h)^2 + k$

$y = a(x - 3)^2 - 5$

Also, assume the function passes through (0,13).

The expression becomes

$13 = a(0 - 3)^2 - 5$

$13 = a(- 3)^2 - 5$

$13 = 9a- 5$

Collect like terms

$9a = 13 + 5$

$9a = 18$

Divide by 9

$a = 2$

Hence, the function is:

$y = a(x - 3)^2 + 5$

$y = 2(x - 3)^2 + 5$

Rewrite as:

$g(x) = (x - 3)^2 + 5$

See attachment for the graph of $g(x) = (x - 3)^2 + 5$

From the graph, the x values span across the x-axis.

This means that the domain of g(x) is: $(-\infty, \infty)$

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