Question

Write each equation in the form y = a(x - h)2 +k. Identify the vertex, focus, directrix, and axis of symmetry of each parabola.

y = -3x2 - 6x + 7

Answer 1

To write the given equation, y = -3x^2 - 6x + 7, in the form y = a(x - h)^2 + k, complete the square and rearrange the equation. The vertex is (-1, 10), the axis of symmetry is x = -1, and the focus and directrix can be determined using the vertex form of the equation.

Step-by-step explanation:

To write the given equation, y = -3x^2 - 6x + 7, in the form y = a(x - h)^2 + k, we need to complete the square. Here are the step-by-step calculations:

1. First, factor out the -3 from the quadratic terms: y = -3(x^2 + 2x) + 7
2. Next, add the square of half the coefficient of x to complete the square: y = -3(x^2 + 2x + 1) + 7 + 3
3. Simplify and rearrange the equation: y = -3(x + 1)^2 + 10

The equation is now in the desired form. The vertex of the parabola is (-1, 10), which corresponds to the values of h and k. The axis of symmetry is x = -1, and the focus and directrix can be determined using the vertex form of the equation.