The parent function is $f(x) = x^2$. The transformation described involves translating the graph of $f(x)$ left by 4 units and down by 2 units.

Step 1: Translation left 4 units

A leftward translation of 4 units changes the input $x$ to $x + 4$. So, the function becomes: $g(x) = (x + 4)^2$

Step 2: Translation down 2 units

A downward translation of 2 units subtracts 2 from the function: $g(x) = (x + 4)^2 - 2$

Step 3: Expand and simplify the equation

Now, expand $(x + 4)^2$: $(x + 4)^2 = x^2 + 8x + 16$ Thus, the function becomes: $g(x) = x^2 + 8x + 16 - 2$ Simplify: $g(x) = x^2 + 8x + 14$

Final equation:

$g(x) = x^2 + 8x + 14$

This is the equation of the transformed function in the form $y = ax^2 + bx + c$.

Do you want more details or have any questions?

Here are 5 related questions to expand on this concept:

1. What would be the equation if the graph were shifted right 3 units and up 5 units?
2. How does a vertical stretch or compression affect the graph's equation?
3. What is the vertex of the function $g(x) = x^2 + 8x + 14$?
4. How would a reflection over the x-axis change the equation of the function?
5. How would you find the x-intercepts of $g(x)$?

Tip: Always expand and simplify the quadratic equation after performing translations to ensure it is in standard form.