Problem 133 - College Algebra

Question

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

Step-by-Step Solution

Answer

To stretch a function's graph vertically, you need to multiply the entire function by a constant factor greater than 1 in its equation. The absolute value of this constant determines how much the function's graph is stretched vertically.

Step 1: Understanding Vertical Stretching

Vertical stretching means that the graph of the function is elongated ('stretched') up and down around its horizontal axis. If the function \( f(x) \) is multiplied by a constant \( a \), where \( |a| > 1 \), then it causes the graph of the function to stretch vertically by a factor of \( a \). For example, if we have a function \( f(x) = x^2 \) and we multiply it by 2, we get a new function \( g(x) = 2x^2 \). The graph of \( g(x) \) is a vertical stretch of the graph of \( f(x) \) by a factor of 2.

Step 2: Applying to a Function's Equation

So, to stretch a function's graph vertically, you have to multiply the entire function by a constant factor greater than 1. In the equation of the function, this looks like \( f(x) = ax \), where \( a \) is the stretching factor. The absolute value of \( a \) (assuming \( a \neq 0 \)) determines how much the function's graph is stretched vertically. For \( |a| > 1 \), the graph is stretched by a factor of \( a \). For \( 0 < |a| < 1 \), the graph is compressed vertically by a factor of \( a \).