Graphs of Quadratic Functions and Transformations

Understanding the Transformation $y = a(x + b)^2 + c$

To understand how the values of $a, b$, and $c$ affect the graph of a quadratic function, we can decompose the expression into a sequence of transformations. We typically assume $a \neq 0$, as $a = 0$ would result in a horizontal straight line. There are multiple logical paths to reach the final equation from the base function $y = x^2$.

Path One: Horizontal Squash and Translations

When $a > 0$, one way to interpret the constant $a$ is as a horizontal transformation. Consider the following sequence:

1. Start with the base function: $y = x^2$.
2. Apply a horizontal scale factor: $y = (\sqrt{a} x)^2 = ax^2$. This represents a horizontal squash towards the y-axis by a scale factor of $1/\sqrt{a}$ if $a > 1$.
3. Apply a horizontal translation: $y = a(x + b)^2$. This moves the graph $b$ units to the left (or $-b$ units in the x-direction).
4. Apply a vertical translation: $y = a(x + b)^2 + c$. This moves the graph $c$ units upwards.

Path Two: Vertical Stretch and Translations

A subtly different and often more intuitive method involves treating $a$ as a vertical scaling factor. This path works for any $a > 0$:

1. Start with the base function: $y = x^2$.
2. Apply a vertical stretch: $y = ax^2$. This is a vertical stretch parallel to the y-axis by a scale factor of $a$.
3. Apply a horizontal translation: $y = a(x + b)^2$. The graph is shifted by $b$ units.
4. Apply a vertical translation: $y = a(x + b)^2 + c$. The graph is shifted by $c$ units.

You can verify that both methods yield the same result. For example, if $a = 4$, Path One involves $y = (2x)^2$, which is a horizontal squash by scale factor $1/2$. Path Two involves $y = 4x^2$, which is a vertical stretch by scale factor $4$. Both result in the same identical curve.

Handling Negative Values of a

When $a$ is negative, such as $a = -4$, we cannot use the square root method from Path One because $\sqrt{-4}$ is not a real number. In such cases, we must include a reflection step. The guide suggests a more detailed sequence:

1. Start with $y = x^2$.
2. Apply a horizontal squash (or vertical stretch) using the absolute value of $a$: $y = |a|x^2$.
3. Reflect the graph in the x-axis: $y = -|a|x^2$, which is equivalent to $y = ax^2$.
4. Apply the horizontal translation: $y = a(x + b)^2$.
5. Apply the vertical translation: $y = a(x + b)^2 + c$.

While this process is more cumbersome, it explains why the parabola opens downwards when $a < 0$. The vertex of the resulting graph is always located at the coordinate $(-b, c)$. To master this, you should pick various sets of values for $a, b$, and $c$ and follow these transformations step by step using a graph sketching tool to see how the curve reacts to each change.