Combinations of Transformations of Parent Functions
Which one of these shows a vertical shift up of 5 units applied to the parent function f(x)=x²
a) f(x) = x²-5
b) f(x) = (x-5)²
c) f(x) = x²+5
d) f(x)=(x+5)²
c) f(x) = x²+5
Question: Which of the following functions has a vertical asymptote at x = 9?
A) f(x) = (1/x) + 9
B) f(x) = (1/x-9)
C) f(x) = √ x-9
D) f(x) = (x-9)2
B, if x = 9, the fraction becomes undefined and a vertical asymptote is created at x = 9.
1. Which parameter value determines a reflection across the x-axis and vertical stretch/compression? A) k
B) a
C) d
D) None of the above
B) a
For the function g(x)=-3(x+2)2+4 Which of the follow statements below true?
a) The graph of the parent function f(x)=x2 has been reflected over the y-axis.
b) The graph has been reflected over the x-axis, vertically stretched by 3, shifted left 2, and shifted up 4.
c) The graph has been horizontally compressed and shifted right 2.
d) The graph is only shifted vertically by 4 units.
b) The graph has been reflected over the x-axis, vertically stretched by 3, shifted left 2, and shifted up 4
If the function f(x) =√(x-h) +k has a domain of (7,∞) and a range of (-4, ∞ ), what is the value of x? Show your reasoning.
(Answer) (Step 1: Identify h and k) Based on domain (7,∞), h = 7. Based on range (-4,∞), k = -4 Therefore, the function becomes f(x) = √(x-7) - 4
(Step 2: Set y to 0) To find x-intercept, set f(x) = 0. 0 = √(x-7) - 4
(Step 3: Solve): 4 = 7
42 = x - 7
16 = x - 7
x = 23 x-intercept is 23
Sarah and Liam are arguing about what the mapping notation formula is for the transformed function
g(x) = 2[5(x - 3)] − 4.
Sarah claims that the correct mapping notation formula is
(x/5 − 3, 2y − 4),
while Liam argues that the correct mapping notation formula is
(x/5 + 3, 2y − 4).
Who is correct and why
Liam is correct. I say this because the horizontal shift in mapping notation is the opposite in sign to the number in the transformed functions formula. Because the transformed function is g(x) = 2[5(x - 3)] − 4, the mapping notation for the x-value is x/4 + 2. Therefore, Liam is correct.
For the following function f(x)=∣x∣ explain how it would be transformed to the function g(x)=−0.5∣x−4∣+3 make sure to include each step and describe how each transformation affects the parent function.
Shifts horizontally by 4 units to the right, the vertex moves from (0,0) to (4,0)
Compressed Vertically by a factor of 0.5, the V shape is now less steep
Reflects about the x-axis, the graph would open downward
Shifts vertically by 3 units upwards, the vertex moves to (4,3)
Overall, the graph would be open downward, be a compressed V, have been shifted right and up from the origin.
Explain the relationship between the domain and range of parent and inverse functions and how they relate. Include an example of a parent function and its inverse.
The relationship between parent and inverse functions are swapped as the inverse undoes the original function. The input (x) and output (y) switch places.
For example: f(x) = x2 + 3
Domain: [0,∞]
Range: [3,∞]
F-1(x)= x-3
Domain: [3,∞]
Range: [0,∞]
The following parent function f(x)=|x| is transformed into g(x) = 6|x-2|-8. Which of the following statements correctly describes the transformations applied to the parent function?
a) Vertical compress BAFO 6, translate right 2, translate up 8.
b) Reflection across the x-axis, vertical stretch BAFO 6, translate left 2 units, translate down 8 units.
c) Vertical stretch BAFO 6, translate right 2 units, translate down 8 units.
d) Vertical stretch BAFO 6, translate left 2 units, translate down 8 units.
c) Vertical stretch BAFO 6, translate right 2 units, translate down 8 units.
Canada's wonderland is designing a water slide, the slide can be represented in the parent f(x)=x2 The ride designers want to:
Flip the ride upside down.
Stretch the slide vertically by a factor of 2
Raise the starting point of the slide 60 meters above the ground.
a) What's the equation of the transformed slide?
b) Find what the height of the slide would be at 1 second into the ride? (in meters)
a) Start with parent
f(x) = x²
Flip upside-down
f(x)= −x²
Vertical stretch by 2
f(x)=−2x²
Raise 60 meters
f(x)=−2x² + 60
Equation:
f(x) = −2x² + 60
b) f(1) = −2(1)² + 60
= 58 meters
Question: A panda is climbing a bamboo stalk. The function H(t) = 2t + 5 represents the panda's height in feet after t minutes.
Rearrange the formula to solve for t.
If the panda is currently 60ft high how many minutes has it been climbing.
If the bamboo stack is currently 95ft high, what is the domain and range of the equation?
Subtract 5 from both sides: H - 5 = 2t
Divide by 2: t = H-52 This is the inverse function
If the panda is currently 60ft high, how many minutes has it been climbing?
t = (60-5)/2
t = (55)/2
Therefore, the panda has been climbing for 27.5 minutes.
If the bamboo stack is currently 95ft high, what is the domain and range of the equation?
Range (Height): The panda starts at t=0, which is H = 2(0)+5 = 5ft. The top is 95ft.
Range: [5, 95]
Domain (Time): 95 = 2t + 5 90 = 2t t = 45.
Domain: [0, 45]
Mario is on a mission to save Princess Peach! The original path he followed is given by the parent function f(x)=x^2. However, Bowser has sent his Goombas to pursue Mario, and Mario must follow the new path defined by g(x) = -6[-3(x + 2)] - 4.
Describe the transformations applied to Mario’s Original path f(x) into the new path.
Reflects across the x-axis
Vertical stretch BAFO 6
Horizontal compression BAFO ⅓
Translate left 2 units
Translate down 4 units