High School

Exercises 69-70 involve Hooke's law, which states that the distance \( x \) that a spring stretches (or compresses) is proportional to the force \( F \) pulling (or pushing) on the spring:

\[ F = kx \]

where \( k \) is the spring constant.

69. \( k = 385.2 \, \text{lb/ft} \)

(a) Graph \( F = kx \).

(b) Write the vertical (or horizontal) stretch or shrink that can be used to transform the graph of \( y = x \) into the graph of \( F = kx \).

Answer :

a) Graph F=kx, where F is the force and x is the distance a spring stretches.

b) To transform the graph of y=x into the graph of F=kx, a vertical stretch or shrink is needed.

a) Graphing F=kx involves plotting the force (F) on the y-axis and the distance (x) the spring stretches on the x-axis. The equation F=kx represents Hooke's law, where k is the spring constant.

The graph will be a straight line passing through the origin with a slope equal to the spring constant (k). As x increases, F will increase proportionally.

b) To transform the graph of y=x into the graph of F=kx, a vertical stretch or shrink is required. The graph of y=x is a straight line with a slope of 1. To match it with F=kx, the graph needs to be vertically stretched or shrunk.

The factor by which the graph should be stretched or shrunk is equal to the spring constant (k). For example, if k=385.2 lb/ft, the graph of y=x should be vertically stretched or shrunk by a factor of 385.2 to obtain the graph of F=kx.

This means that the y-values on the graph will be 385.2 times greater than the corresponding x-values.

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