Given the parabola y + 2 = 2(x + 3)^2,
a. Without using a graphing calculator, determine which direction the parabola opens (up, down, left, or right). Explain your reasoning in a complete sentence(s).
b. Identify the vertex from the equation without using a graphing calculator. Explain your reasoning in a complete sentence(s).
c. Identify the focus algebraically.
d. Identify the directrix algebraically.
You will be graded on finding the correct direction the parabola opens with a correct explanation in a complete sentence, the correct ordered pair for the vertex with a correct explanation in a complete sentence, the correct ordered pair for the focus, correct equation for the directrix, and showing correct, step-by-step work.
a. The parabola opens upwards because the coefficient of the squared term "2(x + 3)^2" is positive, which means it is a vertically oriented parabola.
b. To find the vertex, we can simply rewrite the equation in vertex form by expanding the squared term:
y + 2 = 2(x^2 + 6x + 9)
y + 2 = 2x^2 + 12x + 18
y = 2x^2 + 12x + 16
Now, we can see that the vertex form of a parabola is y = a(x - h)^2 + k, where (h, k) is the vertex. By comparing with the equation above, the vertex is (-3, 16).
c. To find the focus, we use the formula (h, k + 1/4a), where (h, k) is the vertex and "a" is the coefficient of the squared term.
In this case, the vertex is (-3, 16) and "a" is 2. Therefore, the focus is (-3, 16 + 1/(4*2)) = (-3, 16.5).
d. To find the directrix, we use the formula y = k - 1/4a.
Substitute k = 16 and a = 2 into the formula:
y = 16 - 1/(4*2)
y = 16 - 1/8
y = 127/8
Therefore, the directrix is y = 127/8.