Question 1

Find an equation that matches the parabola's graph.&#10;

Accepted Answer

C

Question 2

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Accepted Answer

The given equation is $x^2 = 36y$, which is a parabola opening upwards with its vertex at the origin $(0,0)$. The standard form of a parabola opening upwards is $x^2 = 4py$, where $p$ is the distance from the vertex to the focus and also from the vertex to the directrix. Here, $4p = 36$, so $p = 9$. Therefore, the focus is $(0,9)$ because it is $9$ units above the vertex along the y-axis. The directrix is a line parallel to the x-axis, $9$ units below the vertex, which is $y = -9$. The focal width, which is the distance across the parabola through the focus, is equal to $4p = 36$.

Question 3

Find the standard form of the equation of the parabola.&#10;

Accepted Answer

The standard form of a parabola opening to the right with vertex at the origin is $y^2 = 4px$, where $p$ is the distance from the vertex to the focus. Given the focal width is 6, $4p = 6$, thus $y^2 = 6x$.

Question 4

11eb7f42_bd74_a77f_897e_8955013b6e25_TB8181_00

Accepted Answer

The given equation is in the form $y = ax^2$, where $a = -\frac{1}{40}$. The vertex of a parabola in this form is at the origin $(0,0)$. The focus is at $(0, \frac{1}{4a})$, which calculates to $(0, -10)$ for $a = -\frac{1}{40}$. The directrix is a line parallel to the x-axis, given by $y = -\frac{1}{4a}$, which is $y = 10$ for $a = -\frac{1}{40}$. The focal width, which is the distance across the parabola through the focus, is $|4a|$, which calculates to 40 for $a = -\frac{1}{40}$.

Question 5

Find the standard form of the equation of the parabola.&#10;

Accepted Answer

The parabola has a horizontal axis of symmetry, with the vertex halfway between the focus and directrix. The vertex is at (9,5), and the equation is $(y - 5)^2 = 4(x - 9)$.

Question 6

Find an equation that matches the parabola's graph.&#10;

Accepted Answer

The answer of Find an equation that matches the parabola's...

Question 7

11eb7f42_bd74_ce92_897e_f139e725681f_TB8181_00

Accepted Answer

The equation $y^2 = -12x$ represents a parabola that opens to the left with its vertex at the origin $(0,0)$. The coefficient of $x$ is $-12$, indicating a focal distance of $3$ (since $4p = -12$, thus $p = -3$). The focus is therefore $(-3,0)$, and the directrix is a vertical line $x = 3$. The focal width, given by $4|p|$, is $12$.

Question 8

Find the standard form of the equation of the parabola.&#10;

Accepted Answer

The standard form of the equation for a parabola with a horizontal axis of symmetry and a focus at $(10,0)$ and a directrix at $x = -10$ is $x = \frac{1}{4p}y^2$, where $p$ is the distance from the vertex to the focus or directrix. Here, $p = 10$, so the equation becomes $x = \frac{1}{40}y^2$.

Question 9

Find an equation that matches the parabola's graph.&#10;

Accepted Answer

The answer of Find an equation that matches the parabola's...

Question 10

11eb7f42_bd74_ce90_897e_f3d684485e9e_TB8181_00

Accepted Answer

The equation $x = 8y^2$ is a parabola that opens to the right. The vertex is at the origin $(0,0)$. The standard form is $x = 4py^2$, so $4p = 8$ gives $p = 2$. The focus is $(\frac{1}{32}, 0)$, the directrix is $x = -\frac{1}{32}$, and the focal width is 32.

Question 11

Find an equation that matches the parabola's graph.&#10;

Accepted Answer

The answer of Find an equation that matches the parabola's...

Question 12

11eb7f42_bd74_ce94_897e_c166047eef0a_TB8181_00

Accepted Answer

The answer of 11eb7f42_bd74_ce94_897e_c166047eef0a_TB8181_00...

Question 13

Find an equation that matches the parabola's graph.&#10;

Accepted Answer

The answer of Find an equation that matches the parabola's...

Question 14

Find the standard form of the equation of the parabola.&#10;

Accepted Answer

The answer of Find the standard form of the equation...

Question 15

Find the standard form of the equation of the parabola.&#10;

Accepted Answer

The answer of Find the standard form of the equation...

Question 16

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Accepted Answer

The answer of 11eb7f42_bd74_ce93_897e_a5addcb6167f_TB8181_00...

Question 17

Find an equation that matches the parabola's graph.&#10;

Accepted Answer

The answer of Find an equation that matches the parabola's...

Question 18

Find the standard form of the equation of the parabola.&#10;

Accepted Answer

The answer of Find the standard form of the equation...

Question 19

Find the standard form of the equation of the parabola.&#10;

Accepted Answer

The answer of Find the standard form of the equation...

Question 20

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Accepted Answer

The answer of 11eb7f42_bd74_ce91_897e_6b96274b5b80_TB8181_00...

Question 21

Find the vertex, the focus, and the directrix of the parabola.&#10;

Accepted Answer

The answer of Find the vertex, the focus, and the...

Question 22

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 23

Graph the parabola.&#10;

Accepted Answer

The answer of Graph the parabola.&#10; ...

Question 24

Graph the parabola.&#10;

Accepted Answer

The answer of Graph the parabola.&#10; ...

Question 25

Graph the parabola.&#10;

Accepted Answer

The answer of Graph the parabola.&#10; ...

Question 26

Graph the parabola.&#10;

Accepted Answer

The answer of Graph the parabola.&#10; ...

Question 27

Graph the parabola.&#10;

Accepted Answer

The answer of Graph the parabola.&#10; ...

Question 28

Graph the parabola.&#10;

Accepted Answer

The answer of Graph the parabola.&#10; ...

Question 29

Find the standard form of the equation of the parabola.&#10;

Accepted Answer

The answer of Find the standard form of the equation...

Question 30

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 31

Find the standard form of the equation of the parabola.&#10;

Accepted Answer

The answer of Find the standard form of the equation...

Question 32

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 33

Find the vertex, the focus, and the directrix of the parabola.&#10;

Accepted Answer

The answer of Find the vertex, the focus, and the...

Question 34

Graph the parabola.&#10;

Accepted Answer

The answer of Graph the parabola.&#10; ...

Question 35

Graph the parabola.&#10;

Accepted Answer

The answer of Graph the parabola.&#10; ...

Question 36

Graph the parabola.&#10;

Accepted Answer

The answer of Graph the parabola.&#10; ...

Question 37

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 38

Find the vertex, the focus, and the directrix of the parabola.&#10;

Accepted Answer

The answer of Find the vertex, the focus, and the...

Question 39

Find the standard form of the equation of the parabola.&#10;

Accepted Answer

The answer of Find the standard form of the equation...

Question 40

Graph the parabola.&#10;

Accepted Answer

The answer of Graph the parabola.&#10; ...

Question 41

Graph the ellipse.&#10;

Accepted Answer

The answer of Graph the ellipse.&#10; ...

Question 42

Find an equation in standard form for the ellipse that satisfies the given conditions.&#10;

Accepted Answer

The answer of Find an equation in standard form for...

Question 43

Find the center, vertices, and foci of the ellipse with the given equation.&#10;

Accepted Answer

The answer of Find the center, vertices, and foci of...

Question 44

Find the center, vertices, and foci of the ellipse with the given equation.&#10;

Accepted Answer

The answer of Find the center, vertices, and foci of...

Question 45

Match the given graph with its equation.&#10;

Accepted Answer

The answer of Match the given graph with its equation.&#10;...

Question 46

Match the given graph with its equation.&#10;

Accepted Answer

The answer of Match the given graph with its equation.&#10;...

Question 47

Find the center, vertices, and foci of the ellipse with the given equation.&#10;

Accepted Answer

The answer of Find the center, vertices, and foci of...

Question 48

Find an equation in standard form for the ellipse that satisfies the given conditions.&#10;

Accepted Answer

The answer of Find an equation in standard form for...

Question 49

Find an equation in standard form for the ellipse that satisfies the given conditions.&#10;

Accepted Answer

The answer of Find an equation in standard form for...

Question 50

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 51

Find the center, vertices, and foci of the ellipse with the given equation.&#10;

Accepted Answer

The answer of Find the center, vertices, and foci of...

Question 52

Find the center, vertices, and foci of the ellipse with the given equation.&#10;

Accepted Answer

The answer of Find the center, vertices, and foci of...

Question 53

Match the given graph with its equation.&#10;

Accepted Answer

The answer of Match the given graph with its equation.&#10;...

Question 54

Graph the ellipse.&#10;

Accepted Answer

The answer of Graph the ellipse.&#10; ...

Question 55

Find the center, vertices, and foci of the ellipse with the given equation.&#10;

Accepted Answer

The answer of Find the center, vertices, and foci of...

Question 56

Match the given graph with its equation.&#10;

Accepted Answer

The answer of Match the given graph with its equation.&#10;...

Question 57

Find an equation in standard form for the ellipse that satisfies the given conditions.&#10;

Accepted Answer

The answer of Find an equation in standard form for...

Question 58

Graph the ellipse.&#10;

Accepted Answer

The answer of Graph the ellipse.&#10; ...

Question 59

Match the given graph with its equation.&#10;

Accepted Answer

The answer of Match the given graph with its equation.&#10;...

Question 60

Graph the ellipse.&#10;

Accepted Answer

The answer of Graph the ellipse.&#10; ...

Question 61

Match the given graph with its equation.&#10;

Accepted Answer

The answer of Match the given graph with its equation.&#10;...

Question 62

Find an equation in standard form for the ellipse that satisfies the given conditions.&#10;

Accepted Answer

The answer of Find an equation in standard form for...

Question 63

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 64

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Deck 8: Analytic Geometry in Two and Three Dimensions