- The original function is \(g(x) = 2x^2 + 6\).
- A vertical downward translation by \(6\) units is equivalent to subtracting \(6\) from the entire function.
- The new function is \(h(x) = g(x) - 6\).
- Simplifying gives \(h(x) = (2x^2 + 6) - 6 = 2x^2\), which is our final answer \(\boxed{h(x) = 2x^2}\).
Explanation
- Analysis and Transformation Rule
The problem asks for the expression of a function \(h(x)\) which is obtained by translating the function \(g(x) = 2x^2 + 6\) vertically downward by \(6\) units. A vertical translation of a function \(f(x)\) by \(k\) units is represented by the transformation \(f(x) \to f(x) + k\). If the translation is downward by \(c\) units, we use the transformation \(f(x) \to f(x) - c\). In this specific case, \(c = 6\).
- Calculating the Expression for h(x)
To find \(h(x)\), we apply the vertical downward translation of \(6\) units to \(g(x)\). This means we subtract \(6\) from the entire expression of \(g(x)\):
\[h(x) = g(x) - 6\]
Substituting the definition of \(g(x)\), we get:
\[h(x) = (2x^2 + 6) - 6\]
Now, we simplify this expression:
\[h(x) = 2x^2 + 6 - 6\]
\[h(x) = 2x^2\]
Thus, the expression for the new function is \(h(x) = 2x^2\).
Examples
Vertical transformations are used frequently in fields like engineering and animation. For instance, if you are designing a digital character that needs to move up and down on a screen according to a specific height pattern, you could use a base function like \(g(x) = 2x^2 + 6\) and then translate it downward if the character jumps to a lower platform. Understanding these shifts allows you to manipulate the behavior of systems without needing to redefine the core mathematical relationship governing their movement.