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)+10-(-1/2p+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]
