Question

which graph represents a density curve, and why?
graph a only, because the curve is above the horizontal axis, and the area under the curve is 1
graph b only, because the curve is above the horizontal axis, and the area under the curve is 1
both graph a and graph b, because both curves are above the horizontal axis, and the areas under both curves equal 1
neither graph a nor graph b, because, even though both curves are above the horizontal axis, the area cannot be determined for graph b

Explanation:

Step1: Calculate area for Graph A

For Graph A, it is a rectangle. The formula for the area of a rectangle is \(A = \text{length}\times\text{width}\). The length is \(3-(- 1)=4\), and the width is \(\frac{1}{4}\). So \(A = 4\times\frac{1}{4}=1\).

Step2: Recall density - curve properties

A density curve has two properties: (1) \(y\geq0\) (the curve is above or on the horizontal axis) for all \(x\) in the domain; (2) the area under the curve is \(1\). Graph A satisfies both properties.

Answer:

graph A only, because the curve is above the horizontal axis, and the area under the curve is 1