Problem 133 - Precalculus Essentials

Question

What must be done to a function's equation so that its graph is stretched vertically?

Step-by-Step Solution

Answer

To stretch the graph of a function vertically, you multiply the whole function by a scalar factor \(a > 1\).

Step 1: Start with Original Function

Start by considering any arbitrary function, for example, \(y=f(x)\). This function will have a graph that represents it in the coordinate plane.

Step 2: Apply Vertical Stretch

In order to vertically stretch the graph of a function, we must multiply the entire function by a scalar factor \(a\), where \(a>1\). Thus, the original function \(y=f(x)\) transforms into \( y=a*f(x) \). This will vertically stretch the graph of the original function by a factor of \(a\).

Step 3: Understand the Outcome

Any point \((x, y)\) on the original function's graph will now move to \((x, a*y)\) on the graph of the new function. This means each y-coordinate (height) is multiplied by \(a\), hence, causing a vertical stretch by factor \(a\) in the graph of the function.