Horizontal Translation: Shifting the graph of a function horizontally (left or right) involves altering the input variable. For a function $f(x)$, a horizontal shift is represented as $f(x - h)$, where $h$ determines the direction and magnitude of the shift. If $h > 0$, the graph moves to the right by $h$ units. Conversely, if $h < 0$, the graph shifts to the left by $|h|$ units.

Example: Consider the function $f(x) = x^2$. To shift this graph 3 units to the right, the transformed function becomes $f(x - 3) = (x - 3)^2$. The vertex moves from $(0, 0)$ to $(3, 0)$.

Vertical Translation: Shifting the graph vertically (up or down) involves adding or subtracting a constant to the entire function. For a function $f(x)$, a vertical shift is represented as $f(x) + k$, where $k$ determines the direction and magnitude. If $k > 0$, the graph moves upwards by $k$ units. If $k < 0$, it shifts downwards by $|k|$ units.

Example: Using $f(x) = x^2$ again, to shift the graph 4 units upwards, the transformed function becomes $f(x) + 4 = x^2 + 4$. The vertex moves from $(0, 0)$ to $(0, 4)$.

Reflection over the x-axis: To reflect a function over the x-axis, multiply the entire function by -1. If $f(x)$ is the original function, the reflected function becomes $-f(x)$. This inversion flips the graph vertically.

Example: For $f(x) = x^2$, the reflected function is $-f(x) = -x^2$. The parabola opens downward instead of upward.

Reflection over the y-axis: To reflect a function over the y-axis, replace $x$ with $-x$ in the function. The transformed function is $f(-x)$. This flips the graph horizontally.

Example: With $f(x) = x^2$, the function $f(-x) = (-x)^2 = x^2$ remains unchanged because it is symmetric about the y-axis. However, for a function like $f(x) = x^3$, the reflected function becomes $f(-x) = (-x)^3 = -x^3$, flipping the graph over the y-axis.

Vertical Stretch/Compression: A vertical stretch or compression is achieved by multiplying the function by a scalar. For a function $f(x)$, the transformation is $a \cdot f(x)$, where $a > 1$ causes a vertical stretch (graph becomes taller), and $0 < a < 1$ causes a vertical compression (graph becomes flatter).

Example: Consider $f(x) = x^2$. The function $2f(x) = 2x^2$ stretches the graph vertically, making it steeper. Conversely, $0.5f(x) = 0.5x^2$ compresses the graph vertically.

Horizontal Stretch/Compression: Horizontal stretching or compression involves modifying the input variable by a scaling factor. For a function $f(x)$, the transformation is $f(bx)$, where $b > 1$ results in a horizontal compression (graph becomes narrower), and $0 < b < 1$ causes a horizontal stretch (graph becomes wider).

Example: Using $f(x) = x^2$, the function $f(2x) = (2x)^2 = 4x^2$ compresses the graph horizontally, making it narrower. On the other hand, $f(0.5x) = (0.5x)^2 = 0.25x^2$ stretches the graph horizontally.

Example: Let's transform $f(x) = x^2$ by reflecting it over the x-axis, shifting it 3 units to the right, and stretching it vertically by a factor of 2. The transformed function is:

Step 1: Reflect over the x-axis: $-f(x) = -x^2$

Step 2: Shift 3 units right: $-f(x - 3) = -(x - 3)^2$

Step 3: Vertical stretch by 2: $-2(x - 3)^2$

The final transformed function is $-2(x - 3)^2$, which has its vertex at $(3, 0)$, opens downward, and is steeper than the original parabola.

Function Composition and Transformations: Function transformations can be seen as compositions of basic functions with linear functions representing shifts and scaling factors. For example, horizontal shifts involve modifying the input, while vertical shifts deal with altering the output.

Impact on Function Properties: Each transformation affects specific properties of the function, such as domain, range, symmetry, and intercepts. A systematic analysis is essential to predict how these properties change under various transformations.

Derivation of Horizontal Shifts: Consider a function $f(x)$. To shift it horizontally by $h$ units, we define a new function $g(x) = f(x - h)$. To find the x-coordinate of points on $g(x)$, set $g(x) = f(x - h) = y$. Solving for $x$, we get $x = h + x'$, indicating a rightward shift by $h$ units.

Reflection Proof: Reflecting a function over the x-axis can be represented as $g(x) = -f(x)$. To prove the reflection, consider a point $(a, b)$ on $f(x)$, then the corresponding point on $g(x)$ is $(a, -b)$, effectively mirroring the graph across the x-axis.

Problem 1: Given the function $f(x) = \sqrt{x}$, perform the following transformations sequentially: shift 2 units down, reflect over the y-axis, and horizontally stretch by a factor of 3. Find the equation of the transformed function and sketch its graph.

Solution:

1. Vertical Shift Down by 2 Units: $f(x) - 2 = \sqrt{x} - 2$
2. Reflection over the y-axis: $f(-x) - 2 = \sqrt{-x} - 2$, which is defined for $x \leq 0$.
3. Horizontal Stretch by a factor of 3: $f\left(\frac{-x}{3}\right) - 2 = \sqrt{\frac{-x}{3}} - 2$, defined for $x \leq 0$.

The final transformed function is $g(x) = \sqrt{\frac{-x}{3}} - 2$, which is defined for $x \leq 0$. The graph is a horizontally stretched, reflected, and downward-shifted version of $f(x) = \sqrt{x}$.

Problem 2: A function $h(x)$ is obtained by reflecting the graph of $f(x) = \ln(x)$ over the x-axis, shifting it 4 units to the left, and vertically stretching it by a factor of 3. Determine the domain of $h(x)$.

Solution:

1. Reflection over the x-axis: $-f(x) = -\ln(x)$. Domain: $x > 0$.
2. Horizontal Shift Left by 4 Units: $-f(x + 4) = -\ln(x + 4)$. Domain: $x + 4 > 0 \Rightarrow x > -4$.
3. Vertical Stretch by 3: $-3\ln(x + 4)$. Domain remains $x > -4$.

The domain of $h(x)$ is $x > -4$.

Physics: In kinematics, the position-time graph of an object under uniform acceleration can be analyzed using graph transformations to represent changes in velocity or acceleration.

Engineering: Signal processing often utilizes graph transformations to manipulate and interpret data signals, allowing for filtering, scaling, and shifting of signal waveforms.

Economics: Cost functions in economics can be transformed to reflect changes in production levels or costs, aiding in the analysis of economic models.

• Graph transformations (translation, reflection, stretch) modify the position and shape of function graphs.
• Translations shift graphs horizontally or vertically without altering their shape.
• Reflections flip graphs over the x-axis or y-axis, changing their orientation.
• Stretches and compressions alter the graph's steepness or width.
• Combined transformations allow for complex manipulations of function graphs.