1.12 Transformations of Functions

Transformations! Now, let’s get familiar with additive and multiplicative transformations, which encapsulate translations, dilations, and reflections.

Additive Transformations (Translations)

An additive transformation of a function is a transformation that involves adding or subtracting a constant value to the function.

1️⃣ Vertical Translations

The function $g(x) = f(x) + k$ represents an additive transformation of the function f. In this case, the function f is being shifted vertically by k units. The value of k determines the magnitude and direction of the shift.

The result of this additive transformation is a vertical translation of the graph of f. A vertical translation is a transformation that involves moving the graph of a function up or down along the y-axis. In this case, the graph of f is being moved up or down by k units. ↕️

To visualize the effect of the additive transformation, we can plot the graphs of f and g on the same set of axes. The graph of f would appear in its original position, while the graph of g would be shifted horizontally by k units and vertically by k units.

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2️⃣ Horizontal Translations

The function $g(x) = f(x + h)$ also represents an additive transformation of the function f.

This time, the result of this additive transformation is a horizontal translation of the graph of f. A horizontal translation is a transformation that involves moving the graph of a function left or right along the x-axis. In this case, the graph of f is being moved to the left or right by h units, depending on the sign of h. ↔️

To visualize the effect of the additive transformation, we can plot the graphs of f and g on the same set of axes. The graph of f would appear in its original position, while the graph of g would be shifted horizontally by h units and vertically by 0 units.

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For both cases, it’s crucial to note that the shape of the graph of f remains unchanged by the additive transformation. Only its position on the coordinate plane is altered. 🚨 Therefore, if we know the graph of f, we can easily sketch the graph of g by applying the appropriate horizontal and vertical shifts.

🪞 Multiplicative Transformations (Dilations and Reflections)

A multiplicative transformation involves multiplying the function by a constant value.

1️⃣ Vertical Dilations

The function $g(x) = af(x)$, where a is a non-zero constant, represents a multiplicative transformation of the function f. In this case, the function f is being scaled vertically by a factor of |a|, which means the distance between the function and the x-axis is increased or decreased by a factor of |a|.

The result of this multiplicative transformation is a vertical dilation of the graph of f, a transformation that involves stretching or shrinking the graph of a function vertically.

• If $a > 1$, the dilation causes the graph of f to be stretched vertically, making the curve appear "taller." 🔼
• If $0 < a < 1$, the dilation causes the graph of f to be shrunk vertically, making the curve appear "shorter.” 🔽
• If $a < 0$, the transformation also involves a reflection over the x-axis, which means the graph of f is flipped over the x-axis. 🔁

To visualize the effect of the multiplicative transformation, we can plot the graphs of f and g on the same set of axes. The graph of f would appear in its original position, while the graph of g would be scaled vertically by a factor of |a|.

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2️⃣ Horizontal Dilations

The function $g(x) = f(bx)$, where b is a non-zero constant, also represents a multiplicative transformation of the function f.

In this case, the function f is being scaled horizontally by a factor of $|1/b|$, which means the distance between the function and the y-axis is increased or decreased by a factor of $|1/b|$.

The result of this multiplicative transformation is a horizontal dilation of the graph of f, a transformation that involves stretching or shrinking the graph of a function horizontally.

• If $|b| > 1$, the dilation causes the graph of f to be shrunk horizontally, making the curve appear "wider." ▶️
• If $0 < |b| < 1$, the dilation causes the graph of f to be stretched horizontally, making the curve appear "narrower." ◀️
• If $b < 0$, the transformation also involves a reflection over the y-axis, which means the graph of f is flipped over the y-axis. 🔁

To visualize the effect of the multiplicative transformation, we can plot the graphs of f and g on the same set of axes. The graph of f would appear in its original position, while the graph of g would be scaled horizontally by a factor of $|1/b|$.

Image Courtesy of Nagwa

For both cases, note that the shape of the graph of f remains unchanged by the multiplicative transformation. Only its size is altered. Therefore, if we know the graph of f, we can easily sketch the graph of g by applying the appropriate vertical dilation and reflection, if necessary.

Working with Transformations

Additive and multiplicative transformations can be combined to create more complex transformations of a function. When these transformations are combined, they result in a combination of horizontal and vertical translations and dilations. This means that the graph of the transformed function is shifted both horizontally and vertically and is also scaled horizontally and/or vertically. 😳

For example, consider the function $g(x) = a*f(bx + h) + k$, where a, b, h, and k are constants. This function is a combination of an additive and a multiplicative transformation of the function f. The function f is transformed horizontally by a factor of |1/b| and then horizontally translated by h units. It is also vertically scaled by a factor of a and then vertically translated by k units.

It is important to note that when a function is transformed, its domain and range may change. 😮 The domain of the transformed function may be restricted due to the nature of the transformations.

For example, if a function is reflected over the x-axis, its domain changes from all real numbers to all real numbers except zero. The range of the transformed function may also change due to the vertical scaling.

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