Math 30-1 Translations Textbook Questions
(name)
Describe the translation that has been applied to the graph of f(x) to obtain the graph of g(x).
Determine the equation of the translated function in the form y - k = f(x - h).
Given the graph of y = f(x) and each of the following transformations,
• state the coordinates of the image points A’, B’, C’, D’ and E’.
• sketch the graph of the transformed function
a) g(x) = f(x) + 3
b) h(x) = f(x - 2)
c) s(x) = f(x + 4)
d) t(x) = f(x) – 2
Given the graph of y = f(x), sketch the graph of the transformed function. Describe the
transformation that can be applied to the graph of f(x) to obtain the graph of the transformed
function. Then, write the transformation using mapping notation.
a) r(x) = f(x + 4) - 3
b) s(x) = f(x - 2) - 4
c) t(x) = f(x - 2) + 5
d) v(x) = f(x + 3) + 2
Complete the table.
Translation
Transformed Function
Transformation of Points
vertical
y = f(x) + 5
(x, y) → (x, y + 5)
y = f(x + 7)
(x, y) → (x – 7, y)
y = f(x – 3)
y = f(x) -6
horizontal and vertical
y + 9 = f(x + 4)
(x, y) → (x + 4, y - 6)
horizontal and vertical
(x, y) → (x – 2, y + 3)
horizontal and vertical
y = f(x - h) + k
The graph of the function y = x2 is translated 4 units to the left and 5 units up to form the
transformed function y = g(x).
a) Determine the equation of the function y = g(x).
b) What are the domain and range of the image function?
c) How could you use the description of the translation of the function y = x2 to determine the
domain and range of the image function?
. The graph of f(x) = |x| is transformed to the graph of g(x) = f(x - 9) + 5.
a) Determine the equation of the function g(x).
b) Compare the graph of g(x) to the graph of the base function f(x).
c) Determine three points on the graph of f(x). Write the coordinates of the image points if you
perform the horizontal translation first and then the vertical translation.
d) Using the same original points from part c), write the coordinates of the image points if you
perform the vertical translation first and then the horizontal translation.
e) What do you notice about the coordinates of the image points from parts c) and d)? Is the
order of the translations important?
The graph of the function drawn in red is a translation of the original function drawn in blue.
Write the equation of the translated function in the form y - k = f(x - h).
Use translations to describe how the graph of y = 1/x compares to the graph of each function.
a) y – 4 = 1/x
b) y =
1
x+2
c) y - 3 =
d) y =
1
x−5
1
x+3
–4
a) Given the graph of y = f(x), graph the functions y = -f(x) and y = f(-x).
b) Show the mapping of key points on the graph of y = f(x) to image points on the graphs of y = f(x) and y = f(-x).
c) Describe how the graphs of y = -f(x) and y = f(-x) are related to the graph of y = f(x). State
any invariant points.
Given the graph of y = f(x),
• transform the graph of f(x) to sketch the graph of g(x)
• describe the transformation
• state any invariant points
• state the domain and range of the functions
a) g(x) = 2f(x)
b) g(x) = ½f(x)
c) g(x) = f(2x)
d) g(x) = f(½x)
The graph of the function y = f(x) has been transformed by either a stretch or a reflection. Write
the equation of the transformed graph, g(x).
Given the function y = f(x), write the equation of the form y - k = af(b(x - h)) that would result
from each combination of transformations.
a) a vertical stretch about the x-axis by a factor of 3, a reflection in the x-axis, a horizontal
translation of 4 units to the left, and a vertical translation of 5 units down
b) a horizontal stretch about the y-axis by a factor of 1/3, a vertical stretch about the x-axis by a
factor of ¾, a reflection in both the x-axis and the y-axis, and a translation of 6 units to the right
and 2 units up
. The graph of y = f(x) is given. Sketch the graph of each of the following functions.
a) y + 2 = f(x - 3)
d) y = 3f(1/3x)
b) y = -f(-x)
e) y + 2 = -3f(x + 4)
c) y = f(3(x - 2)) + 1
f) y = ½f(-½(x + 2)) – 1
The graph of the function y = g(x) represents a transformation of the graph of y = f(x).
Determine the equation of g(x) in the form y = af(b(x - h)) + k.
If the x-intercept of the graph of y = f(x) is located at (a, 0) and the y-intercept is located at (0,
b), determine the x and y-intercept after the following transformations of the graph of y = f(x).
a) y = -f(-x)
b) y = 2f(½x)
c) y + 3 = f(x - 4)
d) y + 3 = ½f(¼(x - 4))
Alison often sketches her quilt designs on a coordinate grid. The coordinates for a section of one
her designs are A(-4, 6), B(-2, -2), C(0, 0), D(1, -1), and E(3, 6). She wants to transform the
original design by a horizontal stretch about the y-axis by a factor of 2, a reflection in the x-axis,
and a translation of 4 units up and 3 units to the left.
a) Determine the coordinates of the image points, A’, B’, C’, D’, and E’.
b) If the original design was defined by the function y = f(x), determine the equation of the
design resulting from the transformations.
The graph of the function y = 2x2 + x + 1 is stretched vertically about the x-axis by a factor of 2,
stretched horizontally about the y-axis by a factor of _1 3 , and translated 2 units to the right and
4 units down.
Write the equation of the transformed function.
Consider the function f(x) = x2 - 2.
a) Graph the function f(x). Is the inverse of f(x) a function?
b) Graph the inverse of f(x) on the same set of coordinate axes.
c) Describe how the domain of f(x) could be restricted so that the inverse of f(x) is a function.
Consider the function f(x) = (x + 2)2.
a) Graph the function f(x). Is the inverse of f(x) a function?
b) Graph the inverse of f(x) on the same set of coordinate axes.
c) Describe how the domain of f(x) could be restricted so that the inverse of f(x) is a function.
Algebraically determine the equation of the inverse of each function.
Verify graphically that the relations are inverses of each other.
a) f(x) = 3x + 6
b) f(x) = x2 – 4
Use the reflection line y = x to sketch the graph of x = f(y) on the same set of axes.
State whether or not the graph of the relation is a function. Then, use the horizontal line test to
determine whether the inverse relation will be a function.
Jocelyn and Gerry determine that the inverse of the function f(x) = x2 - 5, x ≥ 0, is f-1(x) =
√𝑥 + 5. Does the graph verify that these functions are inverses of each other? Explain why.
For each of the following functions,
• determine the equation for the inverse, f-1(x)
• graph f(x) and f-1(x)
• determine the domain and range of f(x) and f-1(x)
a) f(x) = 3x + 2
b) f(x) = 4 - 2x
c) f(x) = ½x - 6
d) f(x) = x2 + 2, x ≤ 0
e) f(x) = 2 - x2, x ≥ 0
For each of the following functions,
• determine the equation of the inverse
• graph f(x) and the inverse of f(x)
• restrict the domain of f(x) so that the inverse of f(x) is a function
• with the domain of f(x) restricted, sketch the graphs of f(x) and f-1(x)
a) f(x) = x2 + 3
b) f(x) = ½x2
c) f(x) = -2x2
d) f(x) = │x + 1│
e) f(x) = -(x - 3)2
f) f(x) = │x - 1│- 2
For each function, state two ways to restrict the domain so that the inverse is a function.
a) f(x) = x2 + 4
b) f(x) = 2 - x2
c) f(x) = (x - 3)2
d) f(x) = (x + 2)2 - 4
. Given the function f(x) = 4x - 2, determine each of the following.
a) f-1(4)
b) f-1(-2)
c) f-1(8)
d) f-1(0)
The function for converting the temperature from degrees Fahrenheit, x, to degrees Celsius, y, is
y = 5/9(x - 32).
a) Determine the equivalent temperature in degrees Celsius for 90°F.
b) Determine the inverse of this function. What does it represent? What do the variables
represent?
c) Determine the equivalent temperature in degrees Fahrenheit for 32 °C.
d) Graph both functions. What does the invariant point represent in this situation?