Question

question 4 (5 points)
find the focus and the directrix of the parabola with the equation \\(y = -\frac{1}{12}(x - 4)^2 + 2\\)

a) focus = \\((4, -1)\\), directrix is \\(y = -5\\)
b) focus = \\((4, -1)\\), directrix is \\(y = 5\\)
c) focus = \\((4, 2)\\), directrix is \\(y = 5\\)
d) focus = \\((4, 1)\\), directrix is \\(y = -5\\)

question 5 (5 points)
which function is the result of vertically stretching \\(f(x) = x^2\\) by a factor of 2 and translating it 4 units upward?

a) \\(y = 4x^2 + 2\\)
b) \\(y = 2x^2 + 4\\)
c) \\(y = 2x^2 - 4\\)
d) \\(y = -4x^2 + 2\\)

Explanation:

Identify the vertex of the first parabola

Using the Parabola Vertex Form knowledge point

$$
LATEXBLOCK0
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Calculate the focal parameter and focus

Using the Parabola Focus and Directrix knowledge point

$$
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Determine the directrix of the first parabola

Using the Parabola Focus and Directrix knowledge point

$$
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Apply vertical stretching to the second function

Using the Vertical Stretching knowledge point

$$
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Apply vertical translation to the second function

Using the Vertical Translation knowledge point

$$
LATEXBLOCK4
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Answer:

Question 4

• A) Focus = \((4,-1)\), directrix is \(y = -5\)
• B) Focus = \((4,-1)\), directrix is \(y = 5\) (Correct answer)
• C) Focus = \((4,2)\), directrix is \(y = 5\)
• D) Focus = \((4,1)\), directrix is \(y = -5\)

Question 5

• A) \(y = 4x^2 + 2\)
• B) \(y = 2x^2 + 4\) (Correct answer)
• C) \(y = 2x^2 - 4\)
• D) \(y = -4x^2 + 2\)