4024/22 Oct/Nov 2020, Question 3: Reciprocal Functions

Part (a) · 1 mark

Calculate the missing y-value in the table for y = x/4 + 2/x when x = 7.

1. 1x = 7: y = 7/4 + 2/7 = 1.75 + 0.2857… = 2.0357… ≈ 2.04

Answer2.04 or 2.035 to 2.036

Part (b) · 3 marks

Plot the graph of y = x/4 + 2/x on the grid for x from 0.5 to 7.

1. 1Plot (0.5, 4.13), (1, 2.25), (1.5, 1.71), (2, 1.5), (3, 1.42), (4, 1.5), (5, 1.65), (6, 1.83), (7, 2.04)
2. 2Join with a smooth curve, lowest near x = 3

AnswerCorrect smooth curve

Part (c) · 2 marks

Draw a tangent at x = 1 and use it to estimate the gradient of the curve.

1. 1Draw the tangent to the curve at (1, 2.25)
2. 2Gradient = rise ÷ run of the tangent ≈ −1.75

Answer−2 to −1.1

Part (di) · 2 marks

Draw the straight line 2y + x = 6 on the same grid.

1. 12y + x = 6: when x = 0, y = 3; when y = 0, x = 6
2. 2Rule a straight line through (0, 3) and (6, 0)

AnswerRuled line through (0, 3) and (6, 0)

Part (dii) · 2 marks

State the x-coordinates where the line 2y + x = 6 meets the curve y = x/4 + 2/x.

1. 1Read the x-coordinates where the line crosses the curve
2. 2x ≈ 0.85 and x ≈ 3.15

Answer0.8 to 0.9, 3.1 to 3.2

Part (diii) · 3 marks

Combine 2y + x = 6 with y = x/4 + 2/x to find A and B in 3x² + Ax + B = 0.

1. 1From 2y + x = 6: y = (6 − x)/2
2. 2(6 − x)/2 = x/4 + 2/x
3. 3Multiply by 4x: 12x − 2x² = x² + 8
4. 43x² − 12x + 8 = 0
5. 5A = −12, B = 8

AnswerA = −12, B = 8

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