• Horizontal Translation: Shifting the graph of a function to the left or right involves replacing $x$ with $x - h$, where $h$ is the horizontal shift. The general form is $f(x - h)$.
• Vertical Translation: Shifting the graph upward or downward involves adding or subtracting a constant $k$ to the function. The general form is $f(x) + k$.

Examples:

Consider the basic function $f(x) = x^2$. A horizontal translation by 3 units to the right results in $f(x - 3) = (x - 3)^2$. A vertical translation 4 units upward results in $f(x) + 4 = x^2 + 4$.

• Reflection over the x-axis: Achieved by multiplying the function by $-1$. The transformed function is $-f(x)$.
• Reflection over the y-axis: Achieved by replacing $x$ with $-x$. The transformed function is $f(-x)$.

Examples:

For the function $f(x) = \sqrt{x}$, reflecting over the x-axis gives $-f(x) = -\sqrt{x}$, resulting in the graph appearing below the x-axis. Reflecting over the y-axis transforms it to $f(-x) = \sqrt{-x}$, which reflects the graph across the y-axis.

• Vertical Stretch/Compression: Multiplying the function by a scalar $a$ affects the vertical scale. If $|a| > 1$, the function stretches vertically; if $0 < |a| < 1$, it compresses vertically. The transformed function is $a \cdot f(x)$. For example, $2f(x)$ stretches the graph vertically by a factor of 2.
• Horizontal Stretch/Compression: Replacing $x$ with $bx$ in the function affects the horizontal scale. If $|b| > 1$, the graph compresses horizontally; if $0 < |b| < 1$, it stretches horizontally. The transformed function is $f(bx)$. For instance, $f(2x)$ compresses the graph horizontally by a factor of 2.

Examples:

Take the function $f(x) = \sin(x)$. A vertical stretch by a factor of 3 is represented as $3f(x) = 3\sin(x)$, which increases the amplitude. A horizontal compression by a factor of 2 results in $f(2x) = \sin(2x)$, doubling the frequency.

General Form:

• $a$ represents vertical stretching or compression and reflection over the x-axis if $a$ is negative.
• $b$ represents horizontal stretching or compression and reflection over the y-axis if $b$ is negative.
• $h$ represents horizontal translation.
• $k$ represents vertical translation.

Example:

Apply the transformations: stretch vertically by a factor of 2, compress horizontally by a factor of 3, reflect over the y-axis, translate 4 units right, and 5 units up to the function $f(x) = \sqrt{x}$. The transformed function becomes: $$ 2 \cdot f(-3(x - 4)) + 5 $$ Simplifying, we get: $$ 2 \cdot \sqrt{-3(x - 4)} + 5 $$

Example in Physics:

The position of an object in free fall can be modeled using quadratic functions. Applying vertical and horizontal translations can adjust the model to account for initial height and velocity.

Example of Curriculum Integration:

In IB Mathematics: AA SL, students often encounter questions requiring the identification of transformed functions based on given graphs. Proficiency in recognizing transformations facilitates accurate and efficient problem-solving.

• Function transformations—translation, reflection, stretching, and compression—alter the position, orientation, and shape of graphs.
• Translations shift functions without changing their structure, while reflections create mirror images across axes.
• Stretching and compression modify the scale of functions, affecting their amplitude and frequency.
• Understanding these transformations is crucial for analyzing and interpreting complex functions in various mathematical applications.