- The transformation of \(f(x) = x^2\) to \(g(x) = a \cdot x^2\) is identified as a vertical transformation.
- The constant \(a\) determines the scale and direction of the transformation.
- The resulting function is defined as \(g(x) = a \cdot x^2\).
- Final Answer: Function \(g\) is a \(\boxed{\text{vertical transformation}}\) of function \(f\). \(g(x) = \boxed{a} \cdot x^2\)
Explanation
- Analyzing the Transformation of the Quadratic Function
The function \(f(x) = x^2\) represents a standard parabola with its vertex at the origin. When we transform this function by multiplying it by a constant \(a\), we obtain a new function \(g(x) = a \cdot x^2\). This is known as a vertical transformation of the graph. Depending on the value of \(a\), this transformation results in different behaviors: if \(|a| > 1\), the graph experiences a vertical stretch; if \(0 < |a| < 1\), the graph experiences a vertical compression; and if \(a\) is negative, the graph is reflected across the x-axis in addition to being scaled.
- Defining the General Form of the Transformed Function
A function \(g\) defined by \(g(x) = a \cdot f(x)\) is a vertical transformation of the function \(f\). Given \(f(x) = x^2\), the resulting function is \(g(x) = a \cdot x^2\). Thus, function \(g\) is a vertical transformation (or vertical scaling) of function \(f\), with \(g(x) = a \cdot x^2\) for some constant \(a\).
Examples
Vertical transformations are widely used in fields like physics and engineering to model data. For example, if you are modeling the trajectory of a ball thrown into the air, the height \(h(t)\) can be represented by a quadratic function. If you increase the initial force, you effectively apply a vertical stretch (a scaling factor \(a > 1\)) to the base model \(t^2\), showing how quickly the ball reaches certain heights. Understanding these transformations helps in adjusting models to fit real-world observations accurately.