(a) Complete the table of values for \(y = x^3 - 2x^2 - 3x + 4\)
\(x\)
−2
−1
−0.5
0
1
1.5
2
3
\(y\)
4.875
4
−1.625
(2)
(b) On the grid, draw the graph of \(y = x^3 - 2x^2 - 3x + 4\) for values of \(x\) from −2 to 3(2)
(c) By drawing a suitable straight line on the grid, find estimates for the solutions of the equation \(x^3 - 2x^2 - x + 1 = 0\) Give your solutions correct to 1 decimal place. (4)
Mark scheme (a)
Scheme
Marks
−6, 4, 0, −2, 4
B2
(2)
Notes
B2: Award B1 for 2, 3 or 4 correct.
Mark scheme (b)
Scheme
Marks
correct curve
B2
(2)
Notes
B2: For correct smooth curve. If B2 not awarded, award B1 for at least 5 points plotted correctly ft from table dep on B1 or B2 in (a) (plots ±1 sq)
Mark scheme (c)
Scheme
Marks
\(x^3 - 2x^2 - 3x + 4 = -2x + 3\)
M1
Plot \(y = -2x + 3\)
M1
−0.8 or 0.6 or 2.2
A1
−0.8, 0.6, 2.2
A1
(4)
(8 marks)
Notes
M1: Sufficient to cross curve at least once.
A1: Any one correct \(x\) value at intersection of graphs (or one or more points given as coordinates) ft dep on second M1 (Award even if curve in (a) is incorrect)
A1: Accept −0.9 to −0.7 Accept 0.4 to 0.7 Accept 2.1 to 2.4 (not coordinates) ft (±1 square) dep on second M1 must be 3 values
SC B2 for all correct solutions from graph of \(y = x^3 - 2x^2 - x + 1\)
