Graphs and Transformations: Master the Art of Graphing and Transformations

Question 1

What is the equation of a linear function in slope-intercept form?

Accepted Answer

y = mx + c

Answer

y = ax² + bx + c

Answer

y = x³ + bx² + cx + d

Answer

y = sin(x)

Question 2

What is the gradient of the graph of y = 2x - 1?

Accepted Answer

2

Answer

1

Answer

-1

Answer

-2

Question 3

Which of the following transformations is represented by the equation y = f(-x)?

Accepted Answer

Reflection about the y-axis

Answer

Reflection about the x-axis

Answer

Translation 1 unit to the right

Answer

Translation 1 unit down

Question 4

What is the domain of the function y = 1 / (x - 2)?

Accepted Answer

{x | x ≠ 2}

Answer

{x | x > 2}

Answer

{x | x < 2}

Answer

{x | x ∈ R}

Question 5

What is the range of the function y = |x|?

Accepted Answer

{y | y ≥ 0}

Answer

{y | y < 0}

Answer

{y | y > 0}

Answer

{y | y ∈ R}

Question 6

Find the equation of the linear function with a slope of 2 and a y-intercept of 3.

Accepted Answer

y = 2x + 3

Answer

y = 3x + 2

Answer

y = 2x - 3

Answer

y = 3x - 2

Question 7

Identify the vertex of the parabola y = (x - 2)² + 3.

Accepted Answer

(2, 3)

Answer

(0, 3)

Answer

(2, 0)

Answer

(0, 0)

Question 8

Apply a transformation to translate the graph of y = x² 2 units to the left.

Accepted Answer

y = (x - 2)²

Answer

y = (x + 2)²

Answer

y = x² + 2

Answer

y = x² - 2

Question 9

Specify the domain of the function y = √(x - 3).

Accepted Answer

x ≥ 3

Answer

x ≤ 3

Answer

x > 3

Answer

x < 3

Question 10

Determine the equation of the line that is perpendicular to y = x + 2 and passes through the point (3, 1).

Accepted Answer

y = -x + 4

Answer

y = x - 4

Answer

y = -x + 1

Answer

y = x + 1

Question 11

Identify a characteristic of a logarithmic function.

Accepted Answer

Its inverse function is an exponential function.

Answer

Its graph is an exponential curve.

Answer

Its domain is all positive real numbers.

Answer

Its range is all real numbers.

Question 12

Identify the correct graph for the function f(x) = x^2 + 2x - 3.

Accepted Answer

A parabola with its opening facing upwards

Answer

A parabola with its opening facing downwards

Answer

A linear function with a positive gradient

Answer

A linear function with a negative gradient

Question 13

Identify the transformation applied to the graph of f(x) = x^2 to obtain the graph of g(x) = -x^2 + 3.

Accepted Answer

Reflection over the x-axis followed by a vertical translation upwards

Answer

Reflection over the y-axis followed by a vertical translation downwards

Answer

Reflection over the origin followed by a vertical translation upwards

Answer

Reflection over the origin followed by a vertical translation downwards

Question 14

What is the inverse of the function f(x) = 3x + 2?

Accepted Answer

f^-1(x) = (x - 2) / 3

Answer

f^-1(x) = (x + 2) / 3

Answer

f^-1(x) = 3x - 2

Answer

f^-1(x) = 3x + 2

Question 15

What is the domain of the function f(x) = sqrt(x - 3)?

Accepted Answer

[3, ∞)

Answer

(-∞, 3]

Answer

[0, ∞)

Answer

(-∞, 0]

Question 16

Which transformation would shift the graph of f(x) = x^2 to the left by 2 units?

Accepted Answer

f(x - 2)

Answer

f(x + 2)

Answer

f(-x + 2)

Answer

f(-x - 2)

Question 17

Determine the equation of a linear function with a slope of -2 and a y-intercept of 3.

Accepted Answer

y = -2x + 3

Answer

y = 2x + 3

Answer

y = 3x - 2

Answer

y = -3x + 2

Question 18

A graph of the function f(x) = x^2 is translated 3 units to the right and 2 units up. Write the equation of the transformed function.

Accepted Answer

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

Answer

g(x) = (x - 3)^2 + 2

Answer

g(x) = (x - 3)^2 - 2

Answer

g(x) = (x + 3)^2 - 2

Question 19

Identify a characteristic of the graph of the function y = (x - 2)^3 + 1.

Accepted Answer

All of the characteristics mentioned

Answer

It is a cubic function.

Answer

It has a vertex at (2, 1).

Answer

It is symmetric about the y-axis.

Question 20

Which transformation will result in a narrower graph of the function y = |x|?

Accepted Answer

y = |x/2|

Answer

y = 2|x|

Answer

y = |2x|

Answer

y = 1/2|x|

Question 21

A graph of the function y = e^x is stretched vertically by a factor of 3. Write the equation of the stretched function.

Accepted Answer

g(x) = 3e^x

Answer

g(x) = e^(3x)

Answer

g(x) = 1/3e^x

Answer

g(x) = e^(1/3x)

Question 22

Which of the following functions is odd?

Accepted Answer

y = |x|

Answer

y = x^2

Answer

y = cos(x)

Answer

y = e^x

Question 23

Identify the vertical asymptotes of the graph of the function y = tan(x).

Accepted Answer

x = nπ, where n is an integer

Answer

x = π/2 + nπ, where n is an integer

Answer

x = π/4 + nπ, where n is an integer

Answer

x = 3π/4 + nπ, where n is an integer

Question 24

Which of the following transformations would translate the graph of y = f(x) 3 units to the left?

Accepted Answer

f(x - 3)

Answer

f(x + 3)

Answer

f(3x)

Answer

f(x/3)

Question 25

Find the equation of the line that is perpendicular to the line y = 2x - 1 and passes through the point (1, 3).

Accepted Answer

y = -1/2x + 3.5

Answer

y = -2x + 5

Answer

y = 2x + 1

Answer

y = 1/2x + 2.5

Question 26

Identify the type of graph that represents the function y = |x - 2|.

Accepted Answer

A V-shaped graph with vertex at (2, 0)

Answer

A U-shaped graph with vertex at (0, 2)

Answer

A straight line with gradient 1 and y-intercept -2

Answer

A straight line with gradient -1 and y-intercept 2

Question 27

Which of the following transformations would stretch the graph of y = f(x) vertically by a factor of 2?

Accepted Answer

2f(x)

Answer

f(2x)

Answer

f(x/2)

Answer

f(x) + 2

Question 28

Find the equation of the axis of symmetry of the parabola y = x^2 - 4x + 3.

Accepted Answer

x = 2

Answer

x = 1

Answer

x = 3

Answer

x = 4

Question 29

Determine the inverse of the function f(x) = 2x + 1.

Accepted Answer

f^-1(x) = x/2 - 1/2

Answer

f^-1(x) = 2x - 1

Answer

f^-1(x) = 1/2x + 1/2

Answer

f^-1(x) = -x/2 - 1/2

Question 30

Find the domain of the function y = 1/(x-3).

Accepted Answer

All real numbers except x = 3

Answer

All real numbers greater than 3.

Answer

All real numbers less than 3.

Answer

All real numbers

Question 31

Which transformation moves the graph of f(x) = x^2 one unit to the left?

Accepted Answer

f(x - 1) = x^2

Answer

f(x + 1) = x^2

Answer

f(x) = x^2 + 1

Answer

f(x) = (x - 1)^2 + 1

Question 32

Which transformation reflects the graph of f(x) = x^2 over the y-axis?

Accepted Answer

f(x) = -x^2

Answer

f(-x) = x^2

Answer

f(x) = |x^2|

Answer

f(x) = (x)^2

Question 33

What is the x-coordinate of the vertex of the parabola f(x) = 2(x + 3)^2 - 5?

Accepted Answer

3

Answer

-3

Answer

5

Answer

-5

Question 34

Which transformation stretches the graph of f(x) = x^3 vertically by a factor of 3?

Accepted Answer

f(x) = 3x^3

Answer

f(x) = (1/3)x^3

Answer

f(x) = x^3 + 3

Answer

f(x) = 3(x^3 + 1)

Question 35

What is the domain of the function f(x) = √(x + 2)?

Accepted Answer

x ≥ -2

Answer

x > -2

Answer

x ≥ 2

Answer

x > 2

Question 36

A parabola has the equation (y = x^2 + 2x - 3). Which equation represents the parabola after it has been translated 3 units to the right, and 2 units up?

Accepted Answer

y = (x - 3)^2 + 2(x - 3) - 5

Answer

y = x^2 + 2x - 3

Answer

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

Answer

y = (x + 3)^2 + (x + 3) - 5

Question 37

Which of the following transformations would shift the graph of y = x² vertically upward by 3 units?

Accepted Answer

y = x² + 3

Answer

x = y² + 3

Answer

y = x² - 3

Question 38

What is the equation of the graph obtained by reflecting the graph of y = x³ across the x-axis?

Accepted Answer

y = -x³

Answer

y = -2x³

Answer

y = 2x³

Answer

y = (x - 2)³

Question 39

Which of the following is the equation of the line perpendicular to y = 2x + 1 and passing through the point (-1, 9)?

Accepted Answer

y = (-1/2)x + 8

Answer

y = (1/2)x + 8

Answer

y = 2x - 1

Answer

y = (-1/2)x + 10

Question 40

What is the range of the function f(x) = -|x|?

Accepted Answer

y ≤ 0

Answer

y ≥ 0

Answer

y > 0

Question 41

What is the equation of the line with a gradient of 3 and passing through the point (1, -2)?

Accepted Answer

y = 3x - 5

Answer

y = -3x + 1

Answer

y = -3x - 5

Answer

y = 3x + 1

Question 42

What is the equation of a horizontal line passing through the point (2, 5)?

Accepted Answer

y = 5

Answer

x = 2

Answer

y = 2

Answer

x = 5

Question 43

Which equation represents a cubic function with zeros at x = -1, x = 1, and x = 2?

Accepted Answer

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

Answer

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

Answer

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

Answer

y = (x + 1)(x - 1)(x - 3)

Question 44

What is the equation of a parabola that opens downwards and has its vertex at (0, 3)?

Accepted Answer

y = -x^2 + 3

Answer

y = -x^2 - 3

Answer

y = x^2 + 3

Question 45

What are the coordinates of the vertex of the parabola defined by the equation y = x^2 - 4x + 3?

Accepted Answer

(2, -1)

Answer

(1, -3)

Answer

(0, 3)

Answer

(-2, -1)

Question 46

Which equation represents the graph of y = |x| reflected across the x-axis?

Accepted Answer

y = -|x|

Answer

y = |x - 1|

Answer

y = -x

Answer

y = |x + 1|

Question 47

Which graph represents the function y = 2x^3 - 3x?

Accepted Answer

[image of a cubic function graph]

Answer

[image of a quadratic function graph]

Answer

[image of a linear function graph]

Question 48

What is the slope of a line perpendicular to the line with the equation y = -2x + 5?

Accepted Answer

1/2

Answer

2

Answer

-1/2

Answer

-2

Question 49

Which transformation stretches the graph of y = cos(x) vertically by a factor of 2?

Accepted Answer

y = 2cos(x)

Answer

y = cos(2x)

Answer

y = cos(x/2)

Answer

y = 1/2cos(x)

Question 50

What is the equation of the line tangent to the parabola y = x^2 at the point (1, 1)?

Accepted Answer

y = 2x - 1

Answer

y = -2x + 1

Answer

y = x - 1

Answer

y = x + 1

Question 51

Which transformation results in a graph that is symmetric with respect to the origin?

Accepted Answer

(x, y) -> (-x, -y)

Answer

(x, y) -> (-x, y)

Answer

(x, y) -> (y, x)

Answer

(x, y) -> (x, -y)