Question

Find the equation of the parabola and identify the vertex and focus. Hint: Start with completing the square.

y=8x+2y=−4
a) y=(x−4)2−18, Vertex: (4, -18), Focus: (4, -9)
b) y=(x−4)2+18, Vertex: (4, 18), Focus: (4, 9)
c) y=(x−4)2+18, Vertex: (-4, 18), Focus: (-4, 9)
d) y=(x−4)2−18, Vertex: (-4, -18), Focus: (-4, -9)

Answer 1

The equation of the parabola is y = 8x² + 2x - 4. The vertex is (-1/8, -17/2) and the focus is approximately (-1/8, -271/32).

Step-by-step explanation:

The equation of the parabola is y = 8x² + 2x - 4. To find the vertex and focus, we need to convert the equation into vertex form. Completing the square, we have y = 8(x² + (1/4)x) - 4. Adding and subtracting (1/16)² inside the parentheses, we get y = 8(x² + (1/4)x + (1/16)²) - 4 - 8(1/16)². Simplifying, we have y = 8(x + 1/8)² - 4 - 1/2. Therefore, the equation in vertex form is y = 8(x + 1/8)² - 17/2.

The vertex of the parabola is the point (-1/8, -17/2). To find the focus, we can use the formula c = 1/4a, where a is the coefficient of the squared term. In this case, a = 8, so c = 1/(4*8) = 1/32. The x-coordinate of the focus is the same as the x-coordinate of the vertex, so the focus is (-1/8, -17/2 + 1/32). Simplifying, the focus is approximately (-1/8, -271/32).