Final answer:
The coordinates of the vertex for the equation 2(x+4)²-3=7 are (-4, -3), found by isolating the squared term and using the vertex form of a parabola. Therefore, the coordinates of the vertex are (-4, -3).
Step-by-step explanation:
The equation given by the student is 2(x+4)²-3=7, which is a quadratic equation in vertex form. We can find the vertex of this parabola by solving the equation for x. First, we need to isolate the squared term:
- Add 3 to both sides to get 2(x+4)² = 10.
- Divide both sides by 2 to get (x+4)² = 5.
- Take the square root of both sides which gives x + 4 = ±√5.
- As we are looking for the vertex, we take the positive square root, which gives x = -4 + √5. But since the vertex of a parabola in this form is at the point (h, k) where x = -h, the x-coordinate of the vertex is -4.
- Substitute x back into the equation to find the y-coordinate of the vertex, which yields y = -3 as we get back to the original equation 2(x+4)² - 3 = 7 when x is -4.
Therefore, the coordinates of the vertex are (-4, -3).