Understanding Function Transformations and Mapping Notation

Shape: Vertical Stretch and Compression

The absolute value of $a$ (written as $|a|$) determines how "narrow" or "wide" the parabola appears compared to the standard $y = x^2$ graph:

1. Vertical Stretch ($|a| > 1$): If the value is greater than 1 (or less than -1), the $y$-values are multiplied by a larger number, making the graph appear narrower.
2. Vertical Compression ($0 < |a| < 1$): If the value is a fraction between 0 and 1, the $y$-values are reduced, making the graph appear wider.

Study Tip: To visualize this, think of the $y$-values as being "pulled" away from the x-axis (stretch) or "squashed" toward the x-axis (compression).

Translations: Moving the Graph

While $a$ changes the shape, the parameters $p$ and $q$ are responsible for translations, which move the entire graph without changing its shape or orientation.

• Horizontal Translation ($p$): This moves the graph left or right. In the form $(x - p)^2$, a positive $p$ shifts the graph right, while a negative $p$ (which appears as $x + \text{value}$) shifts the graph left.
• Vertical Translation ($q$): This moves the graph up or down. Adding $q$ shifts the graph up, while subtracting $q$ shifts it down.

Combining Transformations

When a function has multiple transformations applied at once, such as $y = a(x - p)^2 + q$, you must follow a specific order of operations to ensure the graph is drawn correctly.

1. Multiplication First: Apply all stretches, compressions, and reflections (the $a$ parameter).
2. Addition/Subtraction Last: Apply all translations (the $p$ and $q$ parameters).

The complete mapping notation for the function $y = a(x - p)^2 + q$ relative to the parent function $f(x) = x^2$ is: $(x, y) \rightarrow (x + p, ay + q)$ (Note: In this notation, $p$ represents the shift applied to $x$, and $q$ represents the shift applied to $y$).

Worked Example: Analyzing a Complex Function

Let's apply everything we have learned to describe and graph the function: $y = -2(x + 3)^2 + 1$

Step 1: Identify the parameters

• $a = -2$
• $p = -3$ (since the formula is $x - p$, and $x - (-3) = x + 3$)
• $q = 1$

Step 2: Describe the transformations

1. Reflection: Because $a$ is negative, there is a reflection in the x-axis.
2. Vertical Stretch: Because $|a| = 2$, there is a vertical stretch by a factor of 2, making the parabola narrower.
3. Horizontal Translation: The $(x + 3)$ term indicates a shift left by 3 units.
4. Vertical Translation: The $+ 1$ term indicates a shift up by 1 unit.

Step 3: Determine the new vertex The original vertex of $y = x^2$ is $(0, 0)$. Applying our transformations:

• $x: 0 - 3 = -3$
• $y: -2(0) + 1 = 1$ The new vertex is at $(-3, 1)$.

By following these steps, you can transform any parent function into a complex graph with precision.