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Chapter 2: Problem 133

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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 \( a \).

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Most popular questions from this chapter

Explain how to determine if two functions are inverses of each other.

In Tom Stoppard's play Arcadia , the characters dream and talk about mathematics, including ideas involving graphing. composite functions, symmetry, and lack of symmetry in things that are tangled, mysterious, and unpredictable. Group members should read the play. Present a report on the ideas discussed by the characters that are related to concepts that we studied in this chapter. Bring in a copy of the play and read appropriate excerpts.

Use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. In addition, graph the line \(y-x\) and visually determine if \(f\) and g are inverses. $$ f(x)=\frac{1}{x}+2, g(x)=\frac{1}{x-2} $$

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