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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.
Vertical transformations involve changes on the y y y
For function f ( x ) f(x) f ( x ) b b b k k k
Horizontal transformations affect the x x x y y y
For function f ( x ) f(x) f ( x ) y y y a a a
Let's see how combined transformations affect the quadratic function f ( x ) = x 2 f(x) = x^2 f ( x ) = x 2
Initial Function : f ( x ) = x 2 f(x) = x^2 f ( x ) = x 2
Step 1 1 1 - Vertical Translation : Shift 2 units 2 \text{ units} 2 units
Step 2 2 2 - Vertical Dilation : Multiply by factor 0.5 0.5 0.5
Example Calculation for x = 2 x = 2 x = 2
Initial function: f ( 2 ) = 2 2 = 4 f(2) = 2^2 = 4 f ( 2 ) = 2 2 = 4
After translation: f 1 ( 2 ) = 4 + 2 = 6 f_1(2) = 4 + 2 = 6 f 1 ( 2 ) = 4 + 2 = 6
After dilation: g ( 2 ) = 0.5 × 6 = 3 g(2) = 0.5 \times 6 = 3 g ( 2 ) = 0.5 × 6 = 3
If we visualize this, it will look like this:
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
Initial function: f ( x ) = x 2 f(x) = x^2 f ( x ) = x 2
Vertical dilation with factor 2 2 2 h 1 ( x ) = 2 ⋅ f ( x ) = 2 x 2 h_1(x) = 2 \cdot f(x) = 2x^2 h 1 ( x ) = 2 ⋅ f ( x ) = 2 x 2
Vertical translation + 1 +1 + 1 h A ( x ) = h 1 ( x ) + 1 = 2 x 2 + 1 h_A(x) = h_1(x) + 1 = 2x^2 + 1 h A ( x ) = h 1 ( x ) + 1 = 2 x 2 + 1
Order B: Translation first, then dilation
Initial function: f ( x ) = x 2 f(x) = x^2 f ( x ) = x 2
Vertical translation + 1 +1 + 1 h 2 ( x ) = f ( x ) + 1 = x 2 + 1 h_2(x) = f(x) + 1 = x^2 + 1 h 2 ( x ) = f ( x ) + 1 = x 2 + 1
Vertical dilation with factor 2 2 2 h B ( x ) = 2 ⋅ h 2 ( x ) = 2 ( x 2 + 1 ) = 2 x 2 + 2 h_B(x) = 2 \cdot h_2(x) = 2(x^2 + 1) = 2x^2 + 2 h B ( x ) = 2 ⋅ h 2 ( x ) = 2 ( x 2 + 1 ) = 2 x 2 + 2
Example Calculation for x = 1 x = 1 x = 1
Initial function: f ( 1 ) = 1 2 = 1 f(1) = 1^2 = 1 f ( 1 ) = 1 2 = 1
After dilation: h 1 ( 1 ) = 2 × 1 = 2 h_1(1) = 2 \times 1 = 2 h 1 ( 1 ) = 2 × 1 = 2
After translation: h A ( 1 ) = 2 + 1 = 3 h_A(1) = 2 + 1 = 3 h A ( 1 ) = 2 + 1 = 3
Initial function: f ( 1 ) = 1 2 = 1 f(1) = 1^2 = 1 f ( 1 ) = 1 2 = 1
After translation: h 2 ( 1 ) = 1 + 1 = 2 h_2(1) = 1 + 1 = 2 h 2 ( 1 ) = 1 + 1 = 2
After dilation: h B ( 1 ) = 2 × 2 = 4 h_B(1) = 2 \times 2 = 4 h B ( 1 ) = 2 × 2 = 4
It can be seen that the final results are different: h A ( 1 ) = 3 h_A(1) = 3 h A ( 1 ) = 3 h B ( 1 ) = 4 h_B(1) = 4 h B ( 1 ) = 4
For horizontal transformations, we can combine reflection and translation:
Horizontal Transformation Calculation Steps:
Initial Function : f ( x ) = 1.5 x f(x) = 1.5^x f ( x ) = 1. 5 x
Step 1 1 1 - Reflection across y y y : replace x x x − x -x − x
Step 2 2 2 - Horizontal translation : Shift 2 units 2 \text{ units} 2 units
Example Calculation for x = 3 x = 3 x = 3
Initial function: f ( 3 ) = 1.5 3 = 3.375 f(3) = 1.5^3 = 3.375 f ( 3 ) = 1. 5 3 = 3.375
After reflection: f 1 ( 3 ) = 1.5 − 3 = 1 3.375 ≈ 0.296 f_1(3) = 1.5^{-3} = \frac{1}{3.375} \approx 0.296 f 1 ( 3 ) = 1. 5 − 3 = 3.375 1 ≈ 0.296
After translation: g ( 3 ) = 1.5 − ( 3 − 2 ) = 1.5 − 1 = 1 1.5 ≈ 0.667 g(3) = 1.5^{-(3-2)} = 1.5^{-1} = \frac{1}{1.5} \approx 0.667 g ( 3 ) = 1. 5 − ( 3 − 2 ) = 1. 5 − 1 = 1.5 1 ≈ 0.667
Let's visualize this transformation:
Combined transformations have several important properties:
Non-Commutative : The order of transformations affects the final resultCan be Simplified : Some combinations can be written in simpler formsPreserves Continuity : If the original function is continuous, the transformed result is also continuous
Function f ( x ) = x 2 f(x) = x^2 f ( x ) = x 2 3 units 3 \text{ units} 3 units 1 2 \frac{1}{2} 2 1
Function g ( x ) = 2 x g(x) = 2^x g ( x ) = 2 x y y y 1 unit 1 \text{ unit} 1 unit
Compare the transformation results of function h ( x ) = x 2 h(x) = x^2 h ( x ) = x 2
Order A: Vertical dilation factor 3 3 3 2 units 2 \text{ units} 2 units
Order B: Vertical translation 2 units 2 \text{ units} 2 units
Function f ( x ) = x f(x) = \sqrt{x} f ( x ) = x g ( x ) = 2 x + 3 − 1 g(x) = 2\sqrt{x + 3} - 1 g ( x ) = 2 x + 3 − 1
Determine the formula of the transformed function if f ( x ) = ∣ x ∣ f(x) = |x| f ( x ) = ∣ x ∣ 2 units 2 \text{ units} 2 units x x x 3 3 3
Step-by-step transformation of f ( x ) = x 2 f(x) = x^2 f ( x ) = x 2
Step 1 1 1 + 3 +3 + 3
Step 2 2 2 1 2 \frac{1}{2} 2 1
So the formula of the transformed function is g ( x ) = 1 2 x 2 + 3 2 g(x) = \frac{1}{2}x^2 + \frac{3}{2} g ( x ) = 2 1 x 2 + 2 3
Step-by-step transformation of g ( x ) = 2 x g(x) = 2^x g ( x ) = 2 x
Step 1 1 1 y y y
Comparison of two transformation orders:
Order A : Dilation first, then translation
Analysis of transformation g ( x ) = 2 x + 3 − 1 g(x) = 2\sqrt{x + 3} - 1 g ( x ) = 2 x + 3 − 1 f ( x ) = x f(x) = \sqrt{x} f ( x ) = x
Step-by-step transformation of f ( x ) = ∣ x ∣ f(x) = |x| f ( x ) = ∣ x ∣
Step 2 2 2 1 unit 1 \text{ unit} 1 unit
So the formula of the transformed function is h ( x ) = 2 1 − x h(x) = 2^{1-x} h ( x ) = 2 1 − x
Order B : Translation first, then dilation
Different results: h A ( x ) = 3 x 2 + 2 h_A(x) = 3x^2 + 2 h A ( x ) = 3 x 2 + 2 h B ( x ) = 3 x 2 + 6 h_B(x) = 3x^2 + 6 h B ( x ) = 3 x 2 + 6
Transformations that occur sequentially:
Horizontal translation 3 units 3 \text{ units} 3 units f ( x + 3 ) = x + 3 f(x + 3) = \sqrt{x + 3} f ( x + 3 ) = x + 3
Vertical dilation with factor 2 2 2 2 x + 3 2\sqrt{x + 3} 2 x + 3
Vertical translation 1 unit 1 \text{ unit} 1 unit 2 x + 3 − 1 2\sqrt{x + 3} - 1 2 x + 3 − 1
So the formula of the transformed function is g ( x ) = − 3 ∣ x + 2 ∣ g(x) = -3|x + 2| g ( x ) = − 3∣ x + 2∣
