Dilation of Functions in Algebra

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Questions and Answers

What is the result of applying a vertical dilation of 4 and a vertical translation by 6 units to the graph of the function 𝑓?

The y-values are multiplied by 4 and the x-values are unaffected.

The graph is reflected across the x-axis and translated down.

The y-values are only increased by 6.

The y-values are multiplied by 4 and then increased by 6. (correct)

Given the function 𝑔 is defined as 𝑔(𝑥) = 3𝑓(2𝑥) + 𝑏, what effect does the '2' have on the function 𝑓?

It compresses the graph horizontally by a factor of 2. (correct)

It stretches the graph horizontally by a factor of 2.

It flips the graph over the y-axis.

It translates the graph upward by 2 units.

What does finding the zeroes of the function 𝑔 indicate?

The maximum points of the function 𝑔.

The x-values where the graph touches the y-axis.

The x-values where 𝑔(𝑥) = 0. (correct)

The y-values where the graph crosses the x-axis.

If the domain of a function is restricted, which of the following implications might it have?

The function's output values may not cover the entire range. Signup and view all the answers

What represents the y-intercept of the function 𝑔 when evaluated at 𝑔(0)?

The value of g at x=0 directly on the graph. Signup and view all the answers

What effect does the function transformation $g(x) = 2f(x)$ have on the graph of $f(x)$?

It stretches the graph vertically. Signup and view all the answers

If the function $g(x) = f(2x)$ is applied, what transformation is performed on the graph of $f(x)$?

It compresses the graph horizontally. Signup and view all the answers

What is the domain of the transformed function $g(x) = 3f(x) + 2$ if the domain of $f(x)$ is $[2, 5]$?

[2, 5] Signup and view all the answers

For the function $g(x) = f(x) - 3$, how does the range of $g(x)$ compare to the range of $f(x)$ if $f(x)$ has a range of $[4, 9]$?

[1, 6] Signup and view all the answers

If you have a function $f(x)$ with a zero at $x = 1$, what can be said about the function $g(x) = f(3x - 2)$?

It has a zero at $x = 1/3$. Signup and view all the answers

Given the function $f(x)$ has intercepts at $(0, 20)$ and $(2, 0)$, what can be inferred about the function $g(x) = 5f(x)$?

The x-intercept will remain at $(2, 0)$. Signup and view all the answers

If the function $g(x) = f(2x + 1) - 4$, what is the new position of the horizontal shift?

Shift 1 unit to the left. Signup and view all the answers

What transformation is represented by the function $g(x) = f(x - 3) + 5$?

Shift to the right 3 units and then up 5 units. Signup and view all the answers

What is the effect of the transformation $g(x) = 3f(x) + 5$ on the graph of $f(x)$?

It stretches the graph vertically by a factor of 3 and translates it upward by 5 units. Signup and view all the answers

If $g(x) = f(3x) - 1$, what type of transformation does this represent for the function $f(x)$?

Horizontal shrink by a factor of 3 and vertical shift down by 1. Signup and view all the answers

If $g(x) = f(x) + 4$, what is the domain of $g(x)$ if the domain of $f(x)$ is $ ext{[a, b]}$?

The domain of $g(x)$ is the same as $f(x)$, $ ext{[a, b]}$. Signup and view all the answers

Which transformation does $g(x) = f(x - 2) + 3$ represent?

Shift right 2 units and up 3 units. Signup and view all the answers

For which of these functions does the transformation create a vertical reflection?

$g(x) = -f(x)$ Signup and view all the answers

If $f(x)$ has a zero at $x = 3$, what can be said about $g(x) = f(2x - 6)$?

g(x) has a zero at $x = 3$. Signup and view all the answers

What is the range of the function $g(x) = 2f(x) + 1$ if the original range of $f(x)$ is $[c, d]$?

[2c + 1, 2d + 1] Signup and view all the answers

If the domain of $f(x)$ is $[0, 5]$, what is the domain of $g(x) = f(x + 1)$?

[1, 6] Signup and view all the answers

Study Notes

Dilations of Functions

Dilations are multiplicative transformations.
Vertical Dilations:
- Given a graph f(x), if g(x) = af(x), the graph of g(x) is a vertical stretch or compression of f(x) by a factor of |a|. If a > 1, it's a vertical stretch. If 0 < a < 1, it's a vertical compression. If a < 0, there's a vertical stretch or compression, and a reflection over the x-axis.
Horizontal Dilations:
- If g(x) = f(bx), the graph of g(x) is a horizontal stretch or compression of f(x) by a factor of 1/|b|. If b > 1, it's a horizontal compression. If 0 < b < 1, it's a horizontal stretch. If b < 0, there's a horizontal compression or stretch, and a reflection over the y-axis.
Combined Transformations: Transformations can be combined.
- Order matters: Horizontal transformations are applied before vertical transformations in the function.
- Example: g(x) = a f(b(x – c)) + d where "a" affects the vertical scale, "b" affects the horizontal scale, "c" shifts the graph horizontally, and "d" shifts the graph vertically.

Algebraic Transformations

Given a function, find another function based on transformations of the original.
Example: if f(x) = x² – 3x + 2, and g(x) = 2f(x) + 4, what is g(x)?
The transformation of the function is expressed in terms of x.

Numerical Transformations

Transformations of functions based on data represented in a table.
Example: If a table of values for f(x) is given, and g(x) = af(x) +d, evaluate g(x) at a specific point.

Domain and Range Transformations

Given the domain and range of a function, determine the domain and range of a transformation.
Examples: How do horizontal and vertical dilations affect the domain and range?
If f(x) has domain [a, b] and a range [c, d], determine the domain and range of g(x) = af(bx + c) + d.

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Description

Explore the concepts of dilations in functions, including vertical and horizontal transformations. This quiz covers multiplicative behaviors, their impacts on graphs, and the order of transformation applications. Test your understanding of how dilation affects both vertical and horizontal scales in algebra.