A qulliq [kudlik] is an oil lamp that functions as the main source of heat and light for an igloo in the Inuit culture. It is made of soapstone shaped as a parabolic bowl that contains seal blubber for the oil. It is lined with arctic wool for the wick. If the length of the bowl is 30cm and the height is 9cm, determine the quadratic function that can be used to model the shape of a qulliq. Express the function in vertex form.
Graph Q12d
To determine the quadratic function that models the shape of a qulliq, we need to use the vertex form of a quadratic equation:
y = a(x - h)^2 + k,
where (h, k) represents the coordinates of the vertex.
In this case, the bowl is shaped as a parabolic bowl, so we can assume that the parabola opens upwards. The vertex of the parabola will be at the lowest point of the bowl, which will be the deepest point of the qulliq.
Given that the length of the bowl is 30 cm, we can assume that the vertex occurs at x = 15 cm since the vertex is halfway between the points (0, 0) and (30, 0).
Given that the height of the bowl is 9 cm, we can assume that the depth of the qulliq at the vertex is at y = -9 cm.
So, we have the vertex at (h, k) = (15, -9).
Substituting these values into the vertex form of a quadratic equation, we have:
y = a(x - 15)^2 - 9.
Therefore, the quadratic function that models the shape of a qulliq in vertex form is y = a(x - 15)^2 - 9.
To find the quadratic function that can be used to model the shape of a qulliq, we need to determine the vertex form of the quadratic equation.
The vertex form of a quadratic equation is given by: f(x) = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola. In this case, the parabolic bowl of the qulliq can be represented by a quadratic function.
Given that the length of the bowl is 30 cm and the height is 9 cm, we can determine the vertex of the parabola.
The length of the parabolic bowl will correspond to the x-axis, and the height will correspond to the y-axis. Therefore, the x-coordinate of the vertex will be half the length of the bowl, and the y-coordinate will be the height.
Given:
Length of the bowl = 30 cm
Height of the bowl = 9 cm
The x-coordinate of the vertex is:
h = (length of the bowl) / 2 = 30 / 2 = 15 cm
The y-coordinate of the vertex is the same as the height of the bowl:
k = 9 cm
With these values, we can write the quadratic function in vertex form:
f(x) = a(x - h)^2 + k
Substituting the values we found:
f(x) = a(x - 15)^2 + 9
Therefore, the quadratic function that can be used to model the shape of a qulliq in vertex form is:
f(x) = a(x - 15)^2 + 9