Question

consider the function \\(f(x) = x^2\\). which of the following functions shifts \\(f(x)\\) downward 5 units and to the right 3 units?

a) \\(f(x) = (x-5)^2 + 3\\)
b) \\(f(x) = (x-3)^2 - 5\\)
c) \\(f(x) = (x+3)^2 - 5\\)
d) \\(f(x) = (x-5)^2 - 3\\)

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:

Apply horizontal and vertical translations

Using the Horizontal Translation and Vertical Translation knowledge points

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Identify the vertex and parameter p

Using the Parabola Focus and Directrix knowledge point

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Calculate focus and directrix

Using the Parabola Focus and Directrix knowledge point

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Apply vertical stretch and translation

To vertically stretch \(f(x) = x^2\) by a factor of 2, we multiply the function by 2:

$$ y = 2x^2 $$

To translate the function 4 units upward, we add 4 to the expression:

$$ y = 2x^2 + 4 $$

Answer:

Question 1

• (A) \(f(x) = (x-5)^2 + 3\)
• (B) \(f(x) = (x-3)^2 - 5\) (Correct answer)
• (C) \(f(x) = (x+3)^2 - 5\)
• (D) \(f(x) = (x-5)^2 - 3\)

Question 2

• (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 3

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