Quadratic Equations and Functions: Master the Parabola

Question 1

To vertically translate the graph of f(x) = x² - 4x + 3 downward by 2 units, which transformation should be applied?

Accepted Answer

h(x) = x² - 4x - 1

Answer

g(x) = x² - 4x + 5

Answer

k(x) = (x + 2)² - 4x + 3

Answer

l(x) = x² - 2x + 3

Question 2

Find the discriminant of the quadratic equation 2x² + 5x - 3 = 0.

Accepted Answer

19

Answer

-19

Question 3

Find the equation for the axis of symmetry of the parabola with equation y = (x - 1)² + 3.

Accepted Answer

x = 1

Answer

x = -1

Question 4

Which inequality represents the solution set of the quadratic inequality x² - 4x + 3 < 0?

Accepted Answer

x < 1 or x > 3

Answer

x > 1 or x < 3

Question 5

Find the vertex of the parabola represented by the equation y = x² - 4x + 3.

Accepted Answer

(1, -2)

Answer

(1, 0)

Answer

(-1, 0)

Answer

(-1, -2)

Question 6

Determine the axis of symmetry of the parabola given by the equation y = -2(x + 3)² + 1.

Accepted Answer

x = -3

Answer

x = 0

Answer

y = 1

Answer

y = -3

Question 7

Which transformation will shift the graph of y = x² to the right by 3 units?

Accepted Answer

y = (x + 3)²

Answer

y = (x - 3)²

Answer

y = x²(x + 3)

Answer

y = x²(x - 3)

Question 8

Which of the following is the standard form of a quadratic equation?

Accepted Answer

ax² + bx + c = 0

Answer

ax + bx + c = 0

Answer

a(x + b)² + c = 0

Answer

a(x - b)² + c = 0

Question 9

Which of the following transformations shifts a parabola up by 3 units?

Accepted Answer

y = x² + 3

Answer

y = x² - 3

Answer

y = (x - 3)²

Answer

y = (x + 3)²

Question 10

What is the domain of the quadratic function y = √(x - 2)?

Accepted Answer

x ≥ 2

Answer

x ≤ 2

Answer

x > 2

Answer

x < 2

Question 11

Which of the following is the equation of the line of symmetry for the parabola represented by the equation y = 2(x + 1)² - 3?

Accepted Answer

x = 1

Answer

x = -1

Answer

y = -1

Answer

y = 1

Question 12

Find the vertex of the parabola represented by the equation: y = x² - 4x + 3

Accepted Answer

(2, -1)

Answer

(4, -3)

Answer

(2, 1)

Answer

(4, 3)

Question 13

Graphically, a parabola opens upward if the coefficient of x² is:

Accepted Answer

Positive

Answer

Negative

Answer

Zero

Answer

Cannot be determined

Question 14

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

Accepted Answer

Rotation

Answer

Translation

Answer

Reflection

Answer

Stretching

Question 15

What is the equation of the axis of symmetry of the parabola: y = x² + 2x - 3?

Accepted Answer

x = 1

Answer

x = -1

Answer

y = -1

Answer

y = 1

Question 16

Which of the following represents the inverse of the function f(x) = x² - 1?

Accepted Answer

f⁻¹(x) = ±√(x + 1)

Answer

f⁻¹(x) = √(x + 1)

Answer

f⁻¹(x) = -√(x - 1)

Answer

f⁻¹(x) = √(x - 1)

Question 17

Which of the following is a quadratic function?

Accepted Answer

y = x² + 3x - 4

Answer

y = x³ - 2x

Answer

y = √x + 5

Answer

y = sin(x)

Question 18

What is the range of the quadratic function $$f(x) = x^2 + 2x - 1$$?

Accepted Answer

$$[-1, infty)$$

Answer

$$(-infty, -1)$$

Answer

$$[1, infty)$$

Answer

$$(-infty, 1)$$

Question 19

What is the equation of the parabola that has vertex $$(1, 2)$$ and passes through the point $$(3, 10)$$? (Write your answer in vertex form.)

Accepted Answer

$$y = 2(x-1)^2 + 2$$

Answer

$$y = (x-1)^2 + 1$$

Answer

$$y = -2(x-1)^2 + 2$$

Answer

$$y = -(x-1)^2 + 1$$

Question 20

Which of the following is the standard form of a quadratic equation?

Accepted Answer

ax^2 + bx + c = 0

Answer

a(x - h)^2 + k

Answer

y = mx + b

Answer

r = a(cosθ, sinθ)

Question 21

Which of the following is the discriminant of the quadratic equation ax^2 + bx + c = 0?

Accepted Answer

b^2 - 4ac

Answer

2ab - c

Answer

a^2 - 4bc

Answer

4ab + c^2

Question 22

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

Accepted Answer

x = 1

Answer

x = -1

Answer

y = 1

Answer

y = -1

Question 23

Which of the following graphs represents the quadratic function y = -x^2 + 4x - 3?

Accepted Answer

A parabola opening downward

Answer

A parabola opening upward

Answer

A straight line with a positive slope

Answer

A straight line with a negative slope

Question 24

Which of the following transformations can be used to obtain the graph of y = (x - 2)^2 + 3 from the graph of y = x^2?

Accepted Answer

Translation 2 units to the right and 3 units up

Answer

Translation 2 units to the left and 3 units down

Answer

Reflection over the x-axis and translation 2 units to the right

Answer

Reflection over the y-axis and translation 3 units up

Question 25

What is the range of the quadratic function y = -2x^2 + 8x + 3?

Accepted Answer

(∞, -7]

Answer

[-7, ∞)

Answer

[-3, ∞)

Answer

(∞, -3]

Question 26

What is the discriminant of the quadratic equation 2x^2 - 5x + 2 = 0?

Accepted Answer

9

Answer

1

Answer

5

Answer

11

Question 27

MCQ 3: Find the x-intercepts of the parabola y=-x^2+2x+1.

Accepted Answer

(-1,0) and (1,0)

Answer

(0,1)

Answer

(1,0) and (-1,0)

Answer

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

Question 28

MCQ 4: Which quadratic function has a negative leading coefficient?

Accepted Answer

y=-x^2+4x-3

Answer

y=x^2+2x+1

Answer

y=x^2-5x+6

Answer

y=2x^2+3x-1

Question 29

MCQ 6: Which transformation would shift the parabola y=x^2+2x+1 up by 3 units?

Accepted Answer

y=x^2+2x+4

Answer

y=x^2+2x-3

Answer

y=(x+3)^2+1

Answer

y=x^2+(x+3)^2

Question 30

MCQ 9: Which quadratic equation has complex solutions?

Accepted Answer

x^2+6x+13=0

Answer

x^2+2x+1=0

Answer

x^2-4x+5=0

Answer

x^2-2x+2=0

Question 31

Which of the following graphs represents the quadratic function: y = -x² + 2x - 1?

Accepted Answer

A downward opening parabola

Answer

An upward opening parabola

Answer

A straight line

Question 32

What is the equation of the axis of symmetry of the parabola: y = x² - 6x + 5?

Accepted Answer

x = 3

Answer

y = 3

Answer

x = -3

Question 33

What is the minimum value of the quadratic function: y = x² + 4x + 3?

Accepted Answer

-1

Answer

1

Answer

3

Answer

-3

Question 34

Which of the following transformations will shift the graph of y = x² up by 2 units?

Accepted Answer

y = x² + 2

Answer

y = (x - 2)²

Answer

y = (x + 2)²

Answer

y = x² - 2

Question 35

What is the range of the quadratic function: y = -2x² + 4x + 1?

Accepted Answer

(-∞, 1]

Answer

[1, ∞)

Answer

(-∞, ∞)

Answer

(-1, ∞)

Question 36

Which of the following equations represents a quadratic function that has a maximum value of 3 at x = -2?

Accepted Answer

y = -(x + 2)² + 3

Answer

y = -(x - 2)² + 3

Answer

y = (x + 2)² + 3

Question 37

What is the discriminant of the quadratic equation: ax² + bx + c = 0?

Accepted Answer

b² - 4ac

Answer

c² - 4ab

Answer

a² - 4bc

Answer

b² + 4ac

Question 38

The general form of a quadratic equation is?

Accepted Answer

ax² + bx + c = 0

Answer

sin(x) = cos(x)

Answer

mx + b

Answer

x² + y² = r²

Question 39

Find the range of the quadratic function y = -x² + 4x + 1?

Accepted Answer

[-3, ∞)

Answer

(0, 4]

Answer

(-∞, 5]

Answer

(-∞, ∞)

Question 40

Which equation represents a parabola that opens downward?

Accepted Answer

y = -x²

Answer

y = x²

Answer

y = 2x²

Question 41

What is the equation of the axis of symmetry for the parabola defined by the equation y = x² - 4x + 3?

Accepted Answer

x = 2

Answer

y = -2

Answer

y = 2

Answer

x = -2

Question 42

What are the domain and range of the quadratic function: y = (x - 1)² + 2?

Accepted Answer

Domain: All real numbers; Range: [2, ∞)

Answer

Domain: (-∞, 1]; Range: All real numbers

Answer

Domain: All real numbers; Range: (-∞, 2]

Answer

Domain: [1, ∞); Range: All real numbers

Question 43

Solve the quadratic equation by completing the square: x² + 6x - 7 = 0.

Accepted Answer

x = -3 ± √10

Answer

x = 3 ± √10

Answer

x = 3 ± 2√10

Answer

x = -3 ± 2√10

Question 44

Which of the following is an application of quadratic equations in the field of Science?

Accepted Answer

Solving problems involving projectile motion

Answer

Finding the derivative of a polynomial

Answer

Calculating the area of a circle

Answer

Simplifying rational expressions

Question 45

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

Accepted Answer

(-2, -3)

Answer

(-2, 3)

Answer

(2, 3)

Answer

(2, -3)

Question 46

The height, 'h', of a ball thrown vertically upwards is modeled by h(t) = -16t² + 64t + 4, where 't' is the time in seconds. What is the maximum height the ball reaches?

Accepted Answer

68 feet

Answer

16 feet

Answer

4 feet

Answer

64 feet

Question 47

What are the solutions to the quadratic equation x² - 5x + 6 = 0?

Accepted Answer

x = 2 and x = 3

Answer

x = -2 and x = -3

Answer

x = 1 and x = 6

Answer

x = -1 and x = -6

Question 48

Which of the following equations represents a parabola that opens downwards?

Accepted Answer

y = -2x² + 4x - 1

Answer

y = 0.5x² - x + 5

Answer

y = x² + 2x + 3

Answer

y = (x + 1)²

Question 49

What is the discriminant of the quadratic equation 3x² - 2x + 1 = 0?

Accepted Answer

-8

Answer

4

Answer

-4

Answer

16

Question 50

The graph of y = x² is shifted 2 units to the right and 3 units down. What is the equation of the new graph?

Accepted Answer

y = (x - 2)² - 3

Answer

y = (x + 2)² + 3

Answer

y = (x - 2)² + 3

Answer

y = (x + 2)² - 3

Question 51

A rectangular garden has a length that is 3 meters more than twice its width. If the area of the garden is 54 square meters, what is its width?

Accepted Answer

4 meters

Answer

3 meters

Answer

9 meters

Answer

6 meters

Question 52

Which of the following is a factor of the quadratic expression 2x² + 7x + 3?

Accepted Answer

(2x + 1)

Answer

(2x + 3)

Answer

(x + 3)

Answer

(x + 1)

Question 53

What is the sum of the roots of the quadratic equation x^2 - 10x + 21 = 0?

Accepted Answer

10

Answer

-10

Answer

-21

Answer

21

Question 54

Which of the following statements is TRUE about the graph of a quadratic function with a negative discriminant?

Accepted Answer

The graph does not intersect the x-axis.

Answer

The graph is a circle.

Answer

The graph intersects the x-axis at two distinct points.

Answer

The graph is a straight line.