4MA1/2H Jun 2018, Question 14: Solving Equations by Graphical Methods

Part (a) · 2 marks

Fill in the missing y-values in the table for y = x³ − 2x² − 3x + 4 at the given x values.

1. 1x = −2: −8 − 8 + 6 + 4 = −6
2. 2x = −1: −1 − 2 + 3 + 4 = 4
3. 3x = 1: 1 − 2 − 3 + 4 = 0 and x = 2: 8 − 8 − 6 + 4 = −2
4. 4x = 3: 27 − 18 − 9 + 4 = 4, so the values are −6, 4, 0, −2, 4

Answer−6, 4, 0, −2, 4

Part (b) · 2 marks

Draw the curve y = x³ − 2x² − 3x + 4 on the grid for x from −2 to 3.

1. 1Plot the points from part (a): (−2, −6), (−1, 4), (−0.5, 4.875), (0, 4), (1, 0), (1.5, −1.625), (2, −2), (3, 4)
2. 2Draw a smooth curve through all the plotted points
3. 3Ensure the curve passes through each point and has the shape of a cubic graph

AnswerA smooth curve through (−2, −6), (−1, 4), (−0.5, 4.875), (0, 4), (1, 0), (1.5, −1.625), (2, −2), (3, 4)

Part (c) · 4 marks

Draw a suitable straight line and use it to estimate the solutions of x³ − 2x² − x + 1 = 0, to 1 decimal place.

1. 1Rearrange: x³ − 2x² − 3x + 4 = −2x + 3 is the same equation
2. 2So draw the line y = −2x + 3 on the graph
3. 3Read the x-values where the line crosses the curve
4. 4To 1 d.p.: −0.8, 0.6, 2.2

Answer−0.8, 0.6, 2.2

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