question 8 (5 points)
find the equation of the parabola with its focus at $(0,-4)$ and its directrix $y = 4$.
a) $y = -\frac{1}{16}x^2$
b) $y = \frac{1}{16}x^2$
c) $y = -\frac{1}{4}x^2$
d) $y = -\frac{1}{8}x^2$

question 9 (5 points)
which point gives the vertex of $f(x) = -x^2 + 4x - 3$?
a) $(-2,-13)$
b) $(2,-1)$
c) $(2,-13)$
d) $(2,1)$

Question

question 8 (5 points)
find the equation of the parabola with its focus at $(0,-4)$ and its directrix $y = 4$.
a) $y = -\frac{1}{16}x^2$
b) $y = \frac{1}{16}x^2$
c) $y = -\frac{1}{4}x^2$
d) $y = -\frac{1}{8}x^2$

question 9 (5 points)
which point gives the vertex of $f(x) = -x^2 + 4x - 3$?
a) $(-2,-13)$
b) $(2,-1)$
c) $(2,-13)$
d) $(2,1)$

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Parabola Focus and Directrix", "Parabola Vertex Form" ], "new_concepts": [], "current_concepts": [ "Parabola Focus and Directrix", "Parabola Vertex Form", "Quadratic y-intercept" ] } ## Find the vertex of the parabola Using the *Parabola Focus and Directrix* knowledge point $$ \begin{aligned} &\text{Focus: } (h, k + p) = (0, -4)\ &\text{Directrix: } y = k - p = 4\ &k = \frac{-4 + 4}{2} = 0\ &h = 0\ &\text{Vertex: } (h, k) = (0, 0) \end{aligned} $$ ## Determine the value of p and the equation Using the *Parabola Focus and Directrix* knowledge point $$ \begin{aligned} &k + p = -4 \implies 0 + p = -4 \implies p = -4\ &y - k = \frac{1}{4p}(x - h)^2\ &y - 0 = \frac{1}{4(-4)}(x - 0)^2 \implies y = -\frac{1}{16}x^2 \end{aligned} $$ ## Find the x-coordinate of the vertex of f(x) Using the *Parabola Vertex Form* knowledge point $$ \begin{aligned} &f(x) = -x^2 + 4x - 3 \quad (a = -1, b = 4, c = -3)\ &x = -\frac{b}{2a} = -\frac{4}{2(-1)} = 2 \end{aligned} $$ ## Find the y-coordinate of the vertex of f(x) Using the *Parabola Vertex Form* knowledge point $$ \begin{aligned} &y = f(2) = -(2)^2 + 4(2) - 3\ &y = -4 + 8 - 3 = 1\ &\text{Vertex: } (2, 1) \end{aligned} $$ ## Question 8 (A) $y = -1/16x^2$

(B) $y = 1/16x^2$

(C) $y = -1/4x^2$

(D) $y = -1/8x^2$ ## Question 9 (A) $(-2,-13)$

(B) $(2,-1)$

(C) $(2,-13)$

(D) $(2,1)$

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QUESTION IMAGE

Question

Explanation:

Find the vertex of the parabola

Determine the value of p and the equation

Find the x-coordinate of the vertex of f(x)

Find the y-coordinate of the vertex of f(x)

Answer:

Question 8

Question 9