Final answer:
The equation of the quadratic function that goes through the point (2,-4), has an axis of symmetry of -2, and a maximum of 3 is y = -0.25(x+2)^2 + 3.
Step-by-step explanation:
The equation of the quadratic function that goes through the point (2,-4), has an axis of symmetry of -2, and a maximum of 3 can be found using the vertex form of a quadratic function. The vertex form is given by y = a(x-h)^2 + k, where (h,k) is the vertex of the parabola. In this case, the vertex is (-2,3). Plugging in the coordinates of the vertex and the point (2,-4), we can solve for the value of a.
Substituting (-2,3) into the equation, we get 3 = a(2-(-2))^2 + k, which simplifies to 3 = 16a + k. Substituting (2,-4) into the equation, we get -4 = a(2-(-2))^2 + k, which simplifies to -4 = 16a + k. Now we have a system of equations:
3 = 16a + k
-4 = 16a + k
Solving this system of equations, we find that a = -0.25 and k = 3. Substituting these values back into the vertex form equation, we get y = -0.25(x+2)^2 + 3. Therefore, the equation of the quadratic function is y = -0.25(x+2)^2 + 3.