Determine the quadratic function of the following:
Congruent to y=-12x2; opens up; vertex at (-3,2)
since y = -12x^2 has vertex at (0,0) and opens down, your function is reflected in the x-axis, shifted left 3 and up 2, so it is
12(x+3)^2 + 2
To determine the quadratic function congruent to y = -12x^2, opens up, with the vertex at (-3,2), we need to consider the general form of a quadratic function:
y = a(x - h)^2 + k
where "a" is the leading coefficient, and (h, k) represents the vertex.
From the given information, we can determine the values of "a," "h," and "k."
The vertex form of the quadratic function can be written as follows:
y = a(x - (-3))^2 + 2
y = a(x + 3)^2 + 2
To find the value of "a," we can use the fact that the function is congruent to y = -12x^2:
-12x^2 = a(x + 3)^2 + 2
Let's multiply out the right side of the equation:
-12x^2 = a(x^2 + 6x + 9) + 2
-12x^2 = ax^2 + 6ax + 9a + 2
Since the quadratic functions are congruent, the coefficients in front of x^2 should be equal. Hence:
-12 = a
Now we can rewrite the quadratic function with the determined value of "a" and simplify:
y = -12(x + 3)^2 + 2
Therefore, the quadratic function congruent to y = -12x^2, opens up, with a vertex at (-3,2) is:
y = -12(x + 3)^2 + 2
To determine the quadratic function that satisfies the given conditions, we can start with the standard form of a quadratic function:
y = ax^2 + bx + c
Given that the function is congruent to y = -12x^2, we know that the coefficient "a" will remain the same, so a = -12.
Since the quadratic function opens upward, the coefficient "a" must be positive. Therefore, we can rewrite the quadratic function as:
y = 12x^2 + bx + c
Next, we are given that the vertex of the quadratic function is (-3, 2). Using the vertex form of a quadratic function, we can determine the values of "b" and "c".
The vertex form of a quadratic function is:
y = a(x - h)^2 + k
where (h, k) represents the vertex of the quadratic function. In this case, h = -3 and k = 2.
Plugging these values into the vertex form, we get:
y = 12(x - (-3))^2 + 2
y = 12(x + 3)^2 + 2
Therefore, the quadratic function that is congruent to y = -12x^2, opens upward, and has a vertex at (-3, 2) is:
y = 12(x + 3)^2 + 2