Combined Function Transformations

Applying Transformations in Sequence

A combined transformation applies two or more changes to the same function in sequence. Each operation receives the result of the preceding operation, so changing the order can change the final function.

Translation, reflection, rotation, and dilation can be combined. Changing their order can change the final result.

Types of Combined Transformations

Combined transformations split into two families, and each family acts on a different axis. Vertical operations change the output of the function, so they move the graph up, down, or stretch it in height. Horizontal operations change the input, so they move the graph sideways, flip it across the $y$-axis, or change its width.

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

Apply two transformations in sequence to the quadratic function $f(x) = x^2$:

Calculation Steps:

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

2. Step 1 - Vertical Translation: Shift $2 \text{ units}$ up

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

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

   $$\begin{aligned} g(x) &= 0.5 \cdot f_1(x) \\ &= 0.5(x^2 + 2) \\ &= 0.5x^2 + 1 \end{aligned}$$

Example Calculation for $x = 2$:

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

The following graph shows the original function, the translated function, and the dilated function:

Order of Transformations

The order can change the final result. Compare these two sequences:

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: $h_B(1) = 2 \times 2 = 4$

The two transformation orders give different final values: $h_A(1) = 3$ 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 $2 \text{ units}$ to the right

   $$\begin{aligned} g(x) &= f_1(x-2) \\ &= 1.5^{-(x-2)} \\ &= 1.5^{-x+2} \end{aligned}$$

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: $g(3) = 1.5^{-(3-2)} = 1.5^{-1} = \frac{1}{1.5} \approx 0.667$

The graph keeps both intermediate steps visible:

Properties of Combined Transformations

Two combined transformations produce the same graph only when they act on different axes or when they belong to the same family in a compatible order. The table below lists what decides the final result.

The order rule is the one that catches students most often. When two operations act on the same axis, work from the input outward: apply the operation written closest to $x$ first, then move outward.

Exercises

The problems below combine two or more transformations, so read the stated order before writing any formula. Work from the operation closest to $x$ outward, and check whether the two operations act on the same axis. Try them without the answer key first, then compare your steps with the worked answers.

1. Function $f(x) = x^2$ is translated vertically $3 \text{ units}$ 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 $1 \text{ unit}$ to the right. Write the formula of the transformed function.

3. Compare the transformation results of function $h(x) = x^2$ for order A and order B. Order A applies a vertical dilation with factor $3$ first and then a vertical translation of $2 \text{ units}$ up. Order B applies the vertical translation of $2 \text{ units}$ up first and then the vertical dilation with factor $3$. Explain why the two results differ.

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 $2 \text{ units}$ to the left, reflected across the $x$-axis, then dilated vertically with factor $3$.

Answer Key

Every worked answer first rewrites $x$ according to the stated order, then multiplies or adds on the outside. The final graph is checked against the original to confirm that the order was followed.

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.

   GridAutomatic rotation

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 $1 \text{ unit}$ 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.

   GridAutomatic rotation

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.

   GridAutomatic rotation

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

   Transformations that occur sequentially:

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

   GridAutomatic rotation