Question

1. what are the vertices of this ellipse?

options:
(-4, 2) and (4, 2)
(-9, 2) and (-1, 2)
(-5, 5) and (-5, -1)
(2√2, 2) and (2√2, 2)

Explanation:

Step1: Identify ellipse center

First, find the center by averaging the endpoints of the major axis (the longest diameter). The ellipse is horizontal, so use the leftmost and rightmost points: leftmost is $(-8,0)$? No, wait, the ellipse crosses the x-axis at $(-8,0)$ and $(-2,0)$, so center $h = \frac{-8 + (-2)}{2} = -5$, $k = \frac{0 + 4}{2} = 2$ (since the topmost point is $(-5,4)$ and bottom is $(-5,0)$). The major axis is horizontal, so vertices lie on the line $y=2$, at distance $a$ from center. $a = \frac{|-8 - (-2)|}{2} = 3$.

Step2: Calculate vertex coordinates

Vertices are $(h \pm a, k) = (-5 \pm 3, 2)$.
$(-5 + 3, 2) = (-2, 2)$? No, wait, the left vertex is $(-5 - 4, 2) = (-9,2)$? Wait, no, looking at the graph: the leftmost point of the ellipse is $(-9,2)$? No, the grid: each line is 1 unit. The ellipse's left vertex is at $x=-9$, $y=2$, right vertex at $x=-1$, $y=2$. Let's check: center is $(-5,2)$, distance from center to left vertex is $|-5 - (-9)|=4$, to right is $|-5 - (-1)|=4$. Yes, that matches the graph's horizontal span.

Answer:

(-9, 2) and (-1, 2)