Question

(06.02) A linear function that represents the number of animals adopted from the shelter is compared to a different linear function that represents the hours volunteers work at the shelter each week. Describe the key features of the functions that are needed to determine if these lines intersect.

Answer 1

Answer:

1) These linear functions must not describe a pair of parallel lines.

2) Any intersection of two lines determines a point.

Step-by-step explanation:

Hi there!

Well, since no further details were given. Let's call the linear function that represents the number of animals adopted from the shelter $s(x)$. And

the hours the volunteers work at the shelter each week $v(x)$

1) These linear functions must not describe a pair of parallel lines.

If these lines intersect, then their slope must be different.

$m_{s}\neq m_{v}\\e.g.\\m_{s}=2\\m_{v}=-2$

Equal slopes, parallel lines. Parallel lines, no intersection.

Different slopes, then there is an intersection at some point.

2) Any intersection of two lines determines a point. Let's find it.

So to find one common point equalize the two functions, then find the value for the x-coordinate. Plug it in one of the functions, e.g.

$2x+2=-2x+2 \rightarrow 4x=0 \therefore x=0\\2(0)+2=2 \Rightarrow (0,2)$

At the common point, both functions after inserted the same value in the Domain (x) return the same result for their respective  Range set.