Parabolas: Definition, Vertex, Axis of Symmetry, and Equation

Question 1

Which of the following is the standard form of a parabola that opens upwards with its vertex at the origin?

Accepted Answer

y = ax^2 + bx + c

Answer

y = ax^2 - bx + c

Answer

y = -ax^2 - bx + c

Answer

y = -ax^2 + bx + c

Question 2

If the vertex of a parabola is at (2, -1) and it passes through the point (4, 5), determine the equation of the parabola.

Accepted Answer

y = (x - 2)^2 - 1

Answer

y = (x - 2)^2 + 1

Answer

y = (x + 2)^2 - 1

Answer

y = (x + 2)^2 + 1

Question 3

What is the domain of the parabola defined by the equation y = x^2 - 4x + 3?

Accepted Answer

(-∞, ∞)

Answer

[-2, ∞)

Answer

[2, ∞)

Answer

[0, ∞)

Question 4

Which of the following is the range of the parabola y = -2x^2 + 4x + 1?

Accepted Answer

[-1, ∞)

Answer

(1, ∞)

Answer

[0, ∞)

Answer

(-∞, ∞)

Question 5

What is the equation of the line of symmetry of the parabola defined by the equation y = 3(x - 1)^2 - 2?

Accepted Answer

x = 1

Answer

y = -2

Answer

x = -1

Answer

y = 1

Question 6

Which of the following represents the graph of the parabola y = x^2 + 2x + 1?

Accepted Answer

A curve that opens upwards with its vertex at (-1, 0).

Answer

A curve that opens downwards with its vertex at (1, 0).

Answer

A straight line with a slope of 2 and a y-intercept of 1.

Question 7

What is the discriminant of the parabola y = -ax^2 + bx + c?

Accepted Answer

b^2 - 4ac

Answer

a^2 - 4bc

Answer

b^2 + 4ac

Question 8

What is the equation of the axis of symmetry of the parabola with equation y = (x - 3)^2 + 2?

Accepted Answer

x = 3

Answer

y = 3

Answer

y = 2

Answer

x = 2

Question 9

What is the domain of the parabola with equation y = √(x - 2)?

Accepted Answer

[2, ∞)

Answer

(-∞, ∞)

Answer

[0, ∞)

Answer

(-∞, 2]

Question 10

What is the maximum value of the parabola with equation y = -2(x - 1)^2 + 4?

Accepted Answer

4

Answer

2

Answer

-2

Answer

-4

Question 11

Which of the following is the equation of the parabola with vertex at (1, 2) and axis of symmetry x = 1?

Accepted Answer

y = a(x - 1)² + 2

Answer

y = a(x + 1)² + 2

Answer

y = a(x + 2)² + 1

Answer

y = a(x - 2)² + 1

Question 12

Which of the following is the equation of a parabola that opens downward with a vertex at (0, 0)?

Accepted Answer

y = -x²

Answer

y = x²

Answer

y = x

Answer

y = -x

Question 13

A parabola has its vertex at (-1, 4) and passes through the point (1, 0). What is the equation of its axis of symmetry?

Accepted Answer

x = -1

Answer

x = 1

Answer

y = 4

Answer

y = 0

Question 14

Which equation represents a parabola that opens downwards?

Accepted Answer

y = -2(x + 1)² + 5

Answer

y = 2(x - 1)² + 5

Answer

x = -2(y + 1)² + 5

Answer

y = 2(x + 1)² + 5

Question 15

The equation of a parabola is given by y = (x - 3)² - 2. What are the coordinates of its vertex?

Accepted Answer

(3, -2)

Answer

(-3, -2)

Answer

(3, 2)

Answer

(-3, 2)

Question 16

A parabola has a focus at (0, 2) and a directrix at y = -2. What is the distance between the focus and the directrix?

Accepted Answer

4

Answer

sqrt(8)

Answer

2

Answer

8

Question 17

The equation of a parabola is x = -4(y + 1)² + 2. Which way does the parabola open?

Accepted Answer

Left

Answer

Down

Answer

Up

Answer

Right

Question 18

A bridge is designed with a parabolic arch. The arch is 100 meters wide at its base and its highest point is 30 meters above the base. What is the height of the arch 20 meters from the center of the base?

Accepted Answer

25.2 meters

Answer

18 meters

Answer

12 meters

Answer

6 meters

Question 19

Which of the following is a characteristic of all parabolas?

Accepted Answer

They have a directrix and a focus.

Answer

They open upwards.

Answer

They have a vertex at the origin.

Answer

They are symmetric about the x-axis.

Question 20

What is the equation of the parabola with vertex at (2, -1) and focus at (2, 1)?

Accepted Answer

(x - 2)² = 8(y + 1)

Answer

(x - 2)² = -8(y + 1)

Answer

(y + 1)² = 8(x - 2)

Answer

(y + 1)² = -8(x - 2)

Question 21

A parabola has a vertex at (3, 2) and passes through the point (5, 6). What is the equation of the parabola?

Accepted Answer

(x - 3)² = 4(y - 2)

Answer

(y - 2)² = -4(x - 3)

Answer

(x - 3)² = -4(y - 2)

Answer

(y - 2)² = 4(x - 3)

Question 22

What is the axis of symmetry of the parabola defined by the equation (y - 3)² = -12(x + 2)?

Accepted Answer

y = 3

Answer

x = 2

Answer

x = -2

Answer

y = -2

Question 23

Which of the following is NOT a characteristic of a parabola?

Accepted Answer

It has two foci.

Answer

It is defined by a focus and a directrix.

Answer

It is symmetrical.

Answer

It has a vertex.

Question 24

A parabola has a focus at (0, 2) and a directrix at y = -2. What is the equation of the parabola?

Accepted Answer

x² = 8y

Answer

y² = -8x

Answer

y² = 8x

Answer

x² = -8y

Question 25

A parabola opens to the right and has a vertex at the origin. Which of the following is a possible equation for the parabola?

Accepted Answer

y² = 4x

Answer

y² = -4x

Answer

x² = -4y

Answer

x² = 4y

Question 26

A satellite dish is designed in the shape of a parabola. The receiver is located at the focus of the parabola. If the dish is 2 meters wide and 0.5 meters deep, what is the distance between the receiver and the vertex of the dish?

Accepted Answer

0.25 meters

Answer

0.5 meters

Answer

2 meters

Answer

1 meter

Question 27

Which of the following is the equation of a parabola with vertex (0,0) and axis of symmetry y-axis?

Accepted Answer

y = x^2

Answer

y = (x-1)^2

Answer

y = x^2 + 1

Answer

y = 2x^2

Question 28

What is the vertex of the parabola represented by the equation y = 2(x-3)^2 + 5?

Accepted Answer

(3,5)

Answer

(3,0)

Answer

(5,3)

Answer

(0,5)

Question 29

Which of the following is the axis of symmetry of the parabola y = x^2 - 4x + 3?

Accepted Answer

x = 2

Answer

x = 3

Answer

y = -2

Question 30

Find the equation of the parabola that passes through the points (1,4) and (3,2).

Accepted Answer

y = -x^2 + 5x - 2

Answer

y = x^2 + 3x - 2

Answer

y = x^2 + 5x - 2

Answer

y = -x^2 + 3x - 2

Question 31

Which of the following is the discriminant of the quadratic equation y = ax^2 + bx + c?

Accepted Answer

b^2 - 4ac

Answer

4a^2 - b^2c

Answer

b^2 + 4ac

Answer

a^2 - 4bc

Question 32

Which of the following is the equation of a parabola with vertex at (-2, 3) and passing through the point (1, 2)?

Accepted Answer

y = 1/2(x + 2)² + 3

Answer

y = -1/2(x + 2)² + 3

Answer

y = (x - 2)² + 3

Question 33

What is the axis of symmetry of the parabola y = (x - 3)² - 2?

Accepted Answer

x = 3

Answer

y = 2

Answer

y = -2

Answer

x = -3

Question 34

Which of the following graphs represents the parabola y = -x² + 4x - 3?

Accepted Answer

Image of a downward-opening parabola

Answer

Image of a line

Answer

Image of a circle

Answer

Image of an upward-opening parabola

Question 35

What is the vertex of the parabola 2x² - 4x + 3 = 0?

Accepted Answer

(1, -1)

Answer

(-1, 1)

Answer

(2, -1)

Answer

(1, 1)

Question 36

Which of the following is the focus of the parabola y = -1/4(x - 2)² + 1?

Accepted Answer

(2, 2)

Answer

(-2, 1)

Answer

(-2, 2)

Answer

(2, 1)

Question 37

What is the equation of the directrix of the parabola x² = 8y?

Accepted Answer

y = -2

Answer

x = 2

Answer

y = 2

Answer

x = -2

Question 38

Which of the following is a common application of parabolas in the real world?

Accepted Answer

Trajectories of projectiles

Answer

Measuring distances

Answer

Solving quadratic equations

Answer

Approximating functions

Question 39

What is the discriminant of the quadratic equation ax² + bx + c = 0 that determines the nature of the parabola?

Accepted Answer

b² - 4ac

Answer

a² - 4bc

Answer

c² - 4ab

Question 40

Which of the following transformations can be applied to the graph of the parabola y = x² to obtain the graph of y = (x + 2)² - 3?

Accepted Answer

Translation 2 units left and 3 units down

Answer

Reflection across the x-axis and translation 2 units left

Answer

Reflection across the y-axis and translation 3 units down

Answer

Translation 2 units right and 3 units up

Question 41

Consider the parabola y = -0.5x² + 2x + 1. What is the maximum or minimum value of the parabola?

Accepted Answer

Maximum: 2.25

Answer

Minimum: -2.25

Answer

Maximum: 1

Question 42

What is the standard form of the equation of a parabola with vertex at (h, k) and a vertical axis of symmetry?

Accepted Answer

(x - h)² = 4p(y - k)

Answer

(x - h)² = 4p(x - k)

Answer

(y - k)² = 4p(x - h)

Answer

(y - k)² = 4p(y - h)

Question 43

A parabola has a vertex at (-2, 3) and opens to the right. What is the value of 'p' in its standard form equation?

Accepted Answer

p is positive

Answer

p is zero

Answer

p is negative

Answer

p can be positive or negative

Question 44

A parabola has a focus at (2, 5) and a directrix of y = 1. What is the vertex of the parabola?

Accepted Answer

(2, 3)

Answer

(2, 1)

Answer

(2, 5)

Question 45

Find the equation of the parabola with vertex (1, -2) and passing through the point (3, 0).

Accepted Answer

(x - 1)² = 2(y + 2)

Answer

(y + 2)² = 2(x - 1)

Answer

(x - 1)² = -2(y + 2)

Answer

(y + 2)² = -2(x - 1)

Question 46

What is the equation of the axis of symmetry of the parabola (y - 1)² = -8(x + 3)?

Accepted Answer

y = 1

Answer

y = -3

Answer

x = 3

Answer

x = -3

Question 47

A satellite dish is shaped like a parabola. The receiver is placed at the focus of the parabola. If the dish is 4 meters wide and 1 meter deep, what is the distance from the vertex to the receiver?

Accepted Answer

1 meter

Answer

4 meters

Answer

2 meters

Answer

0.5 meters

Question 48

What is the distance between the focus and the directrix of the parabola (x + 2)² = 12(y - 1)?

Accepted Answer

3 units

Answer

12 units

Answer

6 units

Answer

4 units

Question 49

Which of the following is NOT a property of parabolas?

Accepted Answer

All parabolas have a center.

Answer

All parabolas have a directrix.

Answer

All parabolas have an axis of symmetry.

Answer

All parabolas have a focus.

Question 50

A parabola has a vertex at (0, 0) and passes through the point (2, 4). What is the equation of the parabola?

Accepted Answer

x² = 2y

Answer

x² = 4y

Answer

y² = 4x

Answer

y² = 2x

Question 51

A bridge is designed in the shape of a parabolic arch. The arch is 20 meters wide at its base and 5 meters high at its center. What is the equation of the parabola that represents the arch (assuming the vertex is at the origin)?

Accepted Answer

x² = -20y

Answer

y² = -20x

Answer

y² = 20x

Answer

x² = 20y