Unit 5
5 marks · Normal2. The cumulative frequency diagram shows the distribution of times (in minutes) taken by 100 students to finish a puzzle. The median time is read as 42 minutes.
A function representing the time to complete a puzzle is f(x) = (x - 40)^2 + 16.
Explain how the cumulative frequency diagram can be used to estimate the 75th percentile.
Calculate the value of the function f(x) for the median time.
Describe the transformation of the graph y = x^2 to y = f(x).
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
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(a) Find the time at which cumulative frequency equals 75; read this value off the horizontal axis.
(b) f(42) = (42 - 40)^2 + 16 = 4 + 16 = 20
(c) The graph is translated 40 units right and 16 units up relative to y = x^2.
Show mark scheme and worked solution
Marking points
B1 (a) · 1 mark · Correct description of how to read the 75th percentile from cumulative frequency graph
M1 (b) · 1 mark · Substitute median value into the function
A1 (b) · 1 mark · Correct answer for f(42)
B2 (c) · 2 marks · Describes both translations (right, up)
Worked solution
a) The 75th percentile is at 75 students. Draw a horizontal line from 75 on the cumulative frequency axis to the curve, then drop down to the time axis to find the corresponding time.
c) The graph y = x^2 is translated 40 units to the right and 16 units upward to get y = (x - 40)^2 + 16.