6. prove that linear functions grow by equal differences over equal intervals.

part l. this is the graph of y=-1/2x+10 use the graph to show equal intervals of x-values have equal differences of y-values.

a) think about an interval on the axis starting with p + k. what is the difference between x-values? what is the difference between the y-values for these x-values? complete the table using y=-1/2+10.

b) the difference in the x-values for each interval is this value is also constant because =-1/2(p+k)+/2p+10)
=-1/2p-1/2k+10+1/2p-10
=-1/2k

d) does your answer in part c depend on p? if not, what does it depend on?

part ll. use a similar method to prove that for any linear equation, y=mx+b, equal intervals result in growth by equal differences.

step 1: if the equal interval is k, the two endpoints of the interval will be labeled p and p + k. label the endpoints of the corresponding interval on the y-axis of the graph below. hint: first, substitute x=p into y=mx+b. next, substitute x=p+k into y=mx+b

step 2: find the difference between the endpoints of the interval on the y-axis.

(1)
[y-value at x=p]-

(2)
[y-value at x=p+k]

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