Question 1

Which transformation will shift the graph of $$y = \sqrt{x}$$ down by 2 units?

Accepted Answer

$$y = \sqrt{x} - 2$$

Question 2

Given $$f(x) = x^2$, what is the equation of the function after shifting it 3 units to the right and 2 units up?

Accepted Answer

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

Question 3

If $$y = f(x) is $$reflected about the x-axis, which of the following represents the new function?

Accepted Answer

$$y = -f(x)$$

Question 4

Which transformation results in a vertical stretch of the graph of $$y = f(x) by a $$factor of 3?

Accepted Answer

$$y = 3f(x)$$

Question 5

What is the domain and range of $$g(x) = \sqrt{x} - 2$$?

Accepted Answer

Domain: $$[0, \infty)$$, Range: $$[-2, \infty)$$

Question 6

If $$y = f(x)$$, which transformation compresses the graph horizontally by a factor of $$\frac{1}{2}$$?

Accepted Answer

$$y = f(2x)$$

Question 7

A function $$y = f(x) is $$reflected about the y-axis. What is the equation of the new function?

Accepted Answer

$$y = f(-x)$$

Question 8

Given $$y = x^2$, apply the following transformations in order: shift right 3 units, reflect about the x-axis, shift down 2 units. What is the resulting function?

Accepted Answer

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

Question 9

Which of the following describes a horizontal shift 4 units to the left for $$y = f(x)$$?

Accepted Answer

$$y = f(x + 4)$$

Question 10

Suppose $$y = f(x) is $$compressed vertically by a factor of $$\frac{1}{2}$$ and then shifted up by 1 unit. What is the equation of the new function?

Accepted Answer

$$y = \frac{1}{2}f(x) + 1$$

Question 11

A real-world scenario: The height $$h (in $$meters) of a ball thrown upward is modeled by $$h(t) = -5t^2$ + 20t + 2. If $$the entire path is shifted down by 3 meters, what is the new equation?

Accepted Answer

$$h(t) = -5t^2$ + 20t - 1$$

Question 12

Which transformation will result in the graph of $$y = \sqrt{x}$$ being reflected about the x-axis and shifted up by 1 unit?

Accepted Answer

$$y = -\sqrt{x} + 1$$

Question 13

If $$y = f(x) is $$stretched vertically by a factor of 2 and shifted left by 5 units, what is the equation of the new function?

Accepted Answer

$$y = 2f(x + 5)$$

Question 14

Given $$f(x) = x^2$, what is the range of g(x$$) = $$x^2 + 1$?

Accepted Answer

$$[1, \infty)$$

Question 15

Which of the following transformations will result in a horizontal stretch by a factor of 3 for $$y = f(x)$$?

Accepted Answer

$$y = f\left(\frac{x}{3}\right)$$

Question 16

A medical sensor records a signal $$s(t) = \sin(t). If $$the signal is amplified (stretched vertically) by a factor of 4 and shifted down by 2 units, what is the new function?

Accepted Answer

$$s(t) = 4\sin(t) - 2$$

Question 17

Which sequence of transformations will produce $$y = -\sqrt{x + 2} + 1$$ from $$y = \sqrt{x}$$?

Accepted Answer

Shift left 2 units, reflect about the x-axis, shift up 1 unit

Question 18

If $$y = f(x)$$, which transformation will shift the graph 5 units down and reflect it about the y-axis?

Accepted Answer

$$y = f(-x) - 5$$

Question 19

A population model is given by $$P(t) = 1000e$$^{0.02t}$$. If the time variable is compressed horizontally by a factor of 2, what is the new model?

Accepted Answer

$$P(t) = 1000e$$^{0.04t}$$

Graphing Techniques: Transformations of Functions

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