Combined Function Transformations

Understanding Combined Transformations

Combined transformation is the application of two or more transformations sequentially to a function. Imagine cooking with several steps: first we cut vegetables, then sauté, then add spices. Each step changes the basic ingredients into a different form.

In mathematics, we can combine translation, reflection, rotation, and dilation to produce more complex transformations. The order of applying transformations is very important because the final result can be different.

Types of Combined Transformations

Vertical Combined Transformations

Vertical transformations involve changes on the y-axis. A common combination is vertical translation followed by vertical dilation.

For function $f(x)$ that undergoes vertical translation by $b$ then vertical dilation with factor $k$, the formula becomes:

Horizontal Combined Transformations

Horizontal transformations affect the x-axis. An example is reflection across the y-axis followed by horizontal translation.

For function $f(x)$ that is reflected across the y-axis then translated horizontally by $a$, the formula is:

Visualization of Combined Transformations

Let's see how combined transformations affect the quadratic function $f(x) = x^2$:

Calculation Steps:

1. Initial Function: $f(x) = x^2$

2. Step 1 - Vertical Translation: Shift 2 units up

   $$f_1(x) = f(x) + 2 = x^2 + 2$$

3. Step 2 - Vertical Dilation: Multiply by factor 0.5

   $$g(x) = 0.5 \cdot f_1(x) = 0.5(x^2 + 2) = 0.5x^2 + 1$$

Example Calculation for $x = 2$:

• Initial function: $f(2) = 2^2 = 4$
• After translation: $f_1(2) = 4 + 2 = 6$
• After dilation: $g(2) = 0.5 \times 6 = 3$

If we visualize this, it will look like this:

Order of Transformations

The order of applying transformations greatly affects the final result. Let's compare two different orders:

Comparison of Transformation Orders:

Order A: Dilation first, then translation

1. Initial function: $f(x) = x^2$
2. Vertical dilation with factor 2: $h_1(x) = 2 \cdot f(x) = 2x^2$
3. Vertical translation +1: $h_A(x) = h_1(x) + 1 = 2x^2 + 1$

Order B: Translation first, then dilation

1. Initial function: $f(x) = x^2$
2. Vertical translation +1: $h_2(x) = f(x) + 1 = x^2 + 1$
3. Vertical dilation with factor 2: $h_B(x) = 2 \cdot h_2(x) = 2(x^2 + 1) = 2x^2 + 2$

Example Calculation for $x = 1$:

Order A:

• Initial function: $f(1) = 1^2 = 1$
• After dilation: $h_1(1) = 2 \times 1 = 2$
• After translation: $h_A(1) = 2 + 1 = 3$

Order B:

• Initial function: $f(1) = 1^2 = 1$
• After translation: $h_2(1) = 1 + 1 = 2$
• After dilation: $h_B(1) = 2 \times 2 = 4$

It can be seen that the final results are different: $h_A(1) = 3$ while $h_B(1) = 4$.

Combined Horizontal Transformations

For horizontal transformations, we can combine reflection and translation:

Horizontal Transformation Calculation Steps:

1. Initial Function: $f(x) = 1.5^x$

2. Step 1 - Reflection across y-axis: Replace x with -x

   $$f_1(x) = f(-x) = 1.5^{-x}$$

3. Step 2 - Horizontal translation: Shift 2 units to the right

   $$g(x) = f_1(x-2) = 1.5^{-(x-2)} = 1.5^{-x+2}$$

Example Calculation for $x = 3$:

• Initial function: $f(3) = 1.5^3 = 3.375$
• After reflection: $f_1(3) = 1.5^{-3} = \frac{1}{3.375} \approx 0.296$
• After translation: $g(3) = 1.5^{-(3-2)} = 1.5^{-1} = \frac{1}{1.5} \approx 0.667$

Let's visualize this transformation:

Properties of Combined Transformations

Combined transformations have several important properties:

• Non-Commutative: The order of transformations affects the final result
• Can be Simplified: Some combinations can be written in simpler forms
• Preserves Continuity: If the original function is continuous, the transformed result is also continuous

Exercises

1. Function $f(x) = x^2$ is translated vertically 3 units up, then dilated vertically with factor $\frac{1}{2}$. Determine the formula of the transformed function.

2. Function $g(x) = 2^x$ is reflected across the y-axis, then translated horizontally 1 unit to the right. Write the formula of the transformed function.

3. Compare the transformation results of function $h(x) = x^2$ with two different orders:

   • Order A: Vertical dilation factor 3, then vertical translation 2 units up
   • Order B: Vertical translation 2 units up, then vertical dilation factor 3

4. Function $f(x) = \sqrt{x}$ undergoes combined transformations to become $g(x) = 2\sqrt{x + 3} - 1$. List what transformations occur and their order.

5. Determine the formula of the transformed function if $f(x) = |x|$ is translated horizontally 2 units to the left, reflected across the x-axis, then dilated vertically with factor 3.

Answer Key

1. Step-by-step transformation of $f(x) = x^2$:

   Step 1: Vertical translation +3

   $$f_1(x) = x^2 + 3$$

   Step 2: Vertical dilation with factor $\frac{1}{2}$

   $$g(x) = \frac{1}{2}(x^2 + 3) = \frac{1}{2}x^2 + \frac{3}{2}$$

   So the formula of the transformed function is $g(x) = \frac{1}{2}x^2 + \frac{3}{2}$.

   Step-by-Step Transformation: Translation Then Dilation

   Step-by-step transformation of quadratic function.

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2. Step-by-step transformation of $g(x) = 2^x$:

   Step 1: Reflection across y-axis

   $$g_1(x) = 2^{-x}$$

   Step 2: Horizontal translation 1 unit to the right

   $$h(x) = 2^{-(x-1)} = 2^{-x+1} = 2 \cdot 2^{-x}$$

   So the formula of the transformed function is $h(x) = 2^{1-x}$.

   Reflection and Horizontal Translation of Exponential Function

   Transformation of exponential function.

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3. Comparison of two transformation orders:

   Order A: Dilation first, then translation

   $$h_1(x) = 3x^2$$

   $$h_A(x) = 3x^2 + 2$$

   Order B: Translation first, then dilation

   $$h_2(x) = x^2 + 2$$

   $$h_B(x) = 3(x^2 + 2) = 3x^2 + 6$$

   Different results: $h_A(x) = 3x^2 + 2$ and $h_B(x) = 3x^2 + 6$.

   Effect of Transformation Order on Final Result

   Comparison of different transformation orders.

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4. Analysis of transformation $g(x) = 2\sqrt{x + 3} - 1$ from $f(x) = \sqrt{x}$:

   Transformations that occur sequentially:

   • Horizontal translation 3 units to the left: $f(x + 3) = \sqrt{x + 3}$
   • Vertical dilation with factor 2: $2\sqrt{x + 3}$
   • Vertical translation 1 unit down: $2\sqrt{x + 3} - 1$

5. Step-by-step transformation of $f(x) = |x|$:

   $$f_1(x) = |x + 2|$$

   $$f_2(x) = -|x + 2|$$

   $$g(x) = -3|x + 2|$$

   So the formula of the transformed function is $g(x) = -3|x + 2|$.

   Combination of Translation, Reflection, and Dilation of Absolute Function

   Step-by-step transformation of absolute value function.

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