Function Translations Quiz

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Questions and Answers

What sequence of transformations correctly maps the graph of the function 𝑓 to the function 𝑔, where 𝑔(𝑥) = −𝑓(𝑥) + 5?

A vertical translation by 5 units, then a vertical reflection.

A vertical reflection, followed by a vertical translation by 5 units. (correct)

A vertical reflection, followed by a horizontal translation by 5 units.

A horizontal translation by 5 units, then a vertical reflection.

After a vertical translation of the function 𝑓(𝑥) = −𝑥² + 3𝑥 + 2 by 4 units, which function represents the new graph?

𝑚(𝑥) = −(𝑥 + 4)² + 3(𝑥 + 4) + 2

𝑞(𝑥) = −𝑥² + 3𝑥 − 2

𝑝(𝑥) = −𝑥² + 3𝑥 + 6 (correct)

𝑛(𝑥) = −(𝑥 − 4)² + 3(𝑥 − 4) + 2

What effect does the operation of reflecting the function 𝑓 across the x-axis have on its graph?

It preserves the location of the vertex.

It reverses the direction of the graph. (correct)

It changes the concavity of the graph.

It translates the graph horizontally.

Which of the following transformations is needed to convert 𝑓(𝑥) into a function that is vertically translated by 2 units?

Change to 𝑓(𝑥) = −𝑥² + 3𝑥 + 4. Signup and view all the answers

What is the main difference between horizontal and vertical translations of a function?

Horizontal translations affect input values, while vertical translations affect output values. Signup and view all the answers

What is the effect on the graph of the function when it is transformed to $g(x) = f(x) + 4$?

The graph shifts up 4 units. Signup and view all the answers

Given the transformation $g(x) = f(x - 4)$, how does the graph of $g$ relate to the graph of $f$?

The graph shifts to the right by 4 units. Signup and view all the answers

What will be the output of $g(2)$ if $g(x) = f(x) + 2$ and $f(2) = 3$?

5 Signup and view all the answers

If $f(x)$ has a domain of $[-4, 3]$, what will be the domain of $g(x) = -f(x + 5) + 2$?

[-9, -2] Signup and view all the answers

How does the transformation $g(x) = -f(x)$ affect the original graph of $f$?

It reflects the graph across the x-axis. Signup and view all the answers

If $g(x) = f(x - 2) + 3$, how is the range of $g$ compared to the range of $f$?

It is shifted upwards by 3 units. Signup and view all the answers

Given $f(1) = -12$, what is $g(1)$ when $g(x) = f(x) - 4$?

-16 Signup and view all the answers

What transformation is applied to $f(x)$ in the equation $g(x) = -f(x - 2) + 1$?

The graph is reflected and translated. Signup and view all the answers

What is the expression for 𝑔(𝑥) if 𝑓(𝑥) = 4𝑥 + 3?

𝑔(𝑥) = −(4𝑥 + 3) + 5 Signup and view all the answers

If 𝑔(𝑥) is defined as 𝑔(𝑥) = 𝑓(𝑥 − 2) + 5, what does this transformation signify?

Shift the graph of 𝑓 to the right by 2 units and up by 5 units. Signup and view all the answers

For 𝑔(𝑥) = 𝑓(𝑥 + 3) + 4, what is the resulting domain if the original domain of 𝑓 is (0, 5)?

(3, 8) Signup and view all the answers

What does the negative sign in 𝑔(𝑥) = −𝑓(𝑥) + 5 imply about the transformation of the graph of 𝑓?

Reflection over the x-axis, followed by a downward shift. Signup and view all the answers

What is the value of 𝑔(5) if 𝑔(𝑥) = 𝑓(𝑥) + 2 and 𝑓(5) = −2?

0 Signup and view all the answers

If 𝑔(𝑥) = 𝑓(𝑥 − 2) + 4 and 𝑓(1) = 2, what is 𝑔(3)?

6 Signup and view all the answers

If the range of the function 𝑓 is [−4, 8] and 𝑔(𝑥) = −𝑓(𝑥 + 3) + 2, what is the new range of 𝑔(𝑥)?

[−10, 6] Signup and view all the answers

Study Notes

Translations of Functions

Graphic Transformations: Given a graph of a function (f), translations can be applied graphically to create a new graph (g).
Horizontal Shifts: Shifting the graph of f(x) horizontally by 'a' units results in g(x) = f(x - a) (left 'a' units) or g(x) = f(x + a) (right 'a' units).
Vertical Shifts: Shifting the graph of f(x) vertically by 'a' units results in g(x) = f(x) + a (up 'a' units) or g(x) = f(x) - a (down 'a' units).
Vertical Reflection: Reflecting the graph of f(x) across the x-axis creates g(x) = -f(x).

Additive Transformations (Algebraically)

Example 4: Given f(x) = x² - 3x + 2, find g(x) if g(x) = f(x) + 4. (This involves a vertical translation of 4 units up)
Example 4 Alternative (using a table): Given a table of values for f(x), find g(2) if g(x) = f(x) + 2.
Example 4 Alternative (translation): Given a table of values for f(x), find g(y) if g(x) = f(x – 2) + 1.

Numerical Transformations

Example 5: Using a table of values for f(x), find values for g(x) if g(x) = f(x) + 2 or if g(x) = f(x – 2) + 1.

Domain and Range

Example 6: Given the graph of f with a domain of [-4, 3] and a range of (3, 9), find the domain and range of g(x) = -f(x + 5) + 2.
Determining domain and range changes based on how the graph is being transformed.

Practice Problems

Graphics Transformations: Use the graph of f(x) to graph g(x) for various equations involving translations like: g(x) = f(x - 2) + 4, g(x) = f(x + 3), etc.
Algebraic Transformations: Rewrite g(x) in terms of x. Examples: Given f(x) = 4x + 3, find g(x) = f(x) + 5.

Additional Practice (Numeric Transf)

Given a table with x and f(x) values, calculate g(x) for g(x) = f(x) + a, g(x) = f(x + a), other relevant transformations.

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Description

Test your knowledge on graphic transformations and additive transformations of functions. This quiz will cover horizontal and vertical shifts, reflections, and how to derive new functions from given ones. Perfect for students learning about function transformations in mathematics.