Graphs of Functions: Transforming Completed Square Forms

Transforming the Parent Quadratic

To understand how altering the values of $a$, $b$, and $c$ affects the graph of $y = a(x + b)^2 + c$, we must view the expression as a series of transformations applied to the parent function $y = f(x) = x^2$. Each coefficient performs a specific geometric operation on the curve.

Breaking Down the Transformations

There are two primary ways to conceptualise the construction of the graph $y = a(x + b)^2 + c$ from the starting point $y = x^2$. We usually assume $a \neq 0$.

Sequence 1: Horizontal Scaling Approach (for $a > 0$)

1. Start with the parent function: $y = x^2$.
2. Apply a horizontal squash towards the $y$-axis by a factor of $\sqrt{a}$: $y = (\sqrt{a}x)^2 = ax^2$.
3. Translate the graph horizontally by $-b$ units: $y = a(x + b)^2$.
4. Translate the graph vertically by $c$ units: $y = a(x + b)^2 + c$.

Sequence 2: Vertical Scaling Approach (General Case)

1. Start with the parent function: $y = x^2$.
2. Apply a vertical stretch away from the $x$-axis by a scale factor of $a$: $y = ax^2$.
3. Translate the graph horizontally by $-b$ units: $y = a(x + b)^2$.
4. Translate the graph vertically by $c$ units: $y = a(x + b)^2 + c$.

You should verify that both methods produce the same final graph by testing values like $a = 4$. In the first method, we use $f(\sqrt{a}x)$, which is a horizontal squash. In the second, we use $af(x)$, which is a vertical stretch. For the specific case of the quadratic $x^2$, these operations are equivalent.

Dealing with Negative Coefficients

When $a$ is negative, the first sequence becomes more complex because we cannot take the square root of a negative number. In this case, we must introduce an additional reflection step:

1. Start with $y = x^2$.
2. Apply a horizontal squash by factor $\sqrt{|a|}$: $y = |a|x^2$.
3. Reflect the graph in the $x$-axis to account for the negative sign: $y = -|a|x^2$.
4. Apply translations for $b$ and $c$ as before.

While technically correct, this is often cumbersome. It is usually simpler to treat the multiplication by $a$ as a single vertical transformation that includes both a stretch by factor $|a|$ and a reflection in the $x$-axis if $a < 0$.

Geometric Features of the Transformed Graph

By understanding these transformations, we can immediately identify the key features of the quadratic curve without extensive calculation:

1. The Vertex (Turning Point): On the parent graph $y = x^2$, the vertex is at $(0, 0)$. After a horizontal translation by $-b$ and a vertical translation by $c$, the vertex moves to the point $(-b, c)$.
2. The Line of Symmetry: The line of symmetry for $y = x^2$ is $x = 0$. In the transformed graph, this becomes the vertical line $x = -b$.
3. Orientation: If $a > 0$, the graph is a U-shape (concave up). If $a < 0$, the graph is an inverted U-shape (concave down).

You are encouraged to pick different sets of values for $a$, $b$, and $c$ and use a graph sketching package to follow through these transformations step by step. This will help you justify the position and shape of any quadratic curve encountered in the TMUA.