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GCSE · Algebra - Graphs

5.9 Drawing Quadratic Graphs

EDEXCEL 1MA1 · Calculator allowed · about 90 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

The graph of y = x² - 2 is to be drawn for -3 ≤ x ≤ 3.

(a)Complete the table of values for y = x² - 2.
x : -3 , -2 , -1 , 0 , 1 , 2 , 3
y : _ , _ , _ , _ , _ , _ , _(2)

(b)On the grid, plot the points from your table and join them with a smooth curve.(2)

(c)Use your graph to find an estimate for the value of y when x = 1.5.(1)

(d)Use your graph to find estimates for the values of x when y = 5.(2)

(Total for Question 1 is 7 marks)

The graph of y = 4 - x² is to be drawn for -3 ≤ x ≤ 3.

(a)Complete the table of values for y = 4 - x^2.
x : -3 , -2 , -1 , 0 , 1 , 2 , 3
y : _ , _ , _ , _ , _ , _ , _(2)

(b)Draw the graph of y = 4 - x² for -3 ≤ x ≤ 3.(2)

(c)Write down the coordinates of the points where the curve crosses the x-axis.(2)

(d)Write down the coordinates of the turning point of the graph.(1)

(Total for Question 2 is 7 marks)

A rectangular vegetable patch has width x metres and length (x + 3) metres. The area of the patch, A m², is given by A = x(x + 3).

(a)Complete the table of values for A = x(x + 3), for 1 ≤ x ≤ 6.
x : 1 , 2 , 3 , 4 , 5 , 6
A : _ , _ , _ , _ , _ , _(2)

(b)Draw the graph of A = x(x + 3) for 1 ≤ x ≤ 6.(2)

(c)The vegetable patch needs to have an area of 40 m^2. Use your graph to find the value of x.(2)

(Total for Question 3 is 6 marks)

Three students each drew a table of values for y = x² - 3x, for -3 ≤ x ≤ 3. Only one table is correct.

(a)Which table is correct?(1)

A) x: -3,-2,-1,0,1,2,3 -> y: 18,10,4,0,-2,-2,0
B) x: -3,-2,-1,0,1,2,3 -> y: -18,-10,-4,0,2,2,0
C) x: -3,-2,-1,0,1,2,3 -> y: 0,-2,-2,0,4,10,18
D) x: -3,-2,-1,0,1,2,3 -> y: 36,25,16,9,4,1,0

(b)Give a reason for your answer.(1)

(Total for Question 4 is 2 marks)

For each equation, state whether its graph is u-shaped or n-shaped.

(i)y = x² - 5(1)

(ii)y = 7 - 2x²(1)

(iii)y = 3x² + x - 1(1)

(iv)y = 1 - x - x²(1)

(Total for Question 5 is 4 marks)

The graph of y = x² + 2x - 3 is to be drawn for -4 ≤ x ≤ 2.

(a)Complete the table of values for y = x² + 2x - 3.
x : -4 , -3 , -2 , -1 , 0 , 1 , 2
y : _ , _ , _ , _ , _ , _ , _(2)

(b)Draw the graph of y = x² + 2x - 3 for -4 ≤ x ≤ 2.(2)

(c)Use the graph to write down the solutions of x² + 2x - 3 = 0.(2)

(d)Write down the coordinates of the minimum point of the graph.(1)

(Total for Question 6 is 7 marks)

A ball is thrown in the air. Its height, h metres, above the ground after t seconds is given by h = 20t - 5t^2.

(a)Complete the table of values for h = 20t - 5t^2.
t : 0 , 0.5 , 1 , 1.5 , 2 , 2.5 , 3 , 3.5 , 4
h : 0 , 8.75 , 15 , _ , 20 , _ , 15 , _ , 0(3)

(b)Draw the graph of h = 20t - 5t² for 0 ≤ t ≤ 4.(2)

(c)Use the graph to find the maximum height reached by the ball.(1)

(d)Use the graph to estimate the times at which the ball is at a height of 12 metres.(2)

(Total for Question 7 is 8 marks)

The grid shows an accurately drawn graph of y = x² - x - 6 for -3 ≤ x ≤ 4.

(a)Use the graph to write down the solutions of x² - x - 6 = 0.(2)

(b)By drawing the line y = 4 on the grid, use the graphs to find estimates for the solutions of x² - x - 6 = 4.(3)

(Total for Question 8 is 5 marks)

The equation of a graph is y = -x² + 4.

(a)State whether the graph is u-shaped or n-shaped, giving a reason.(2)

(b)Write down the coordinates of the y-intercept of the graph.(1)

(c)Write down the coordinates of the maximum point of the graph.(1)

(d)Which of these is the correct set of x-axis intercepts for y = -x² + 4?(1)

A) (-4, 0) and (4, 0)
B) (-2, 0) and (2, 0)
C) (-16, 0) and (16, 0)
D) no x-axis intercepts

(Total for Question 9 is 5 marks)

The graph of y = 3x² - 5 is to be drawn for -2 ≤ x ≤ 2.

(a)Complete the table of values for y = 3x² - 5.
x : -2 , -1 , -0.5 , 0 , 0.5 , 1 , 2
y : 7 , -2 , _ , -5 , _ , -2 , 7(2)

(b)Draw the graph of y = 3x² - 5 for -2 ≤ x ≤ 2.(2)

(c)Use your graph to find estimates for the solutions of 3x² - 5 = 0.(2)

(Total for Question 10 is 6 marks)

The graph of y = x² + 3x + 5 is a u-shaped curve that lies entirely above the x-axis; it does not cross or touch the x-axis at any point.

(a)Explain what this tells you about the solutions of the equation x² + 3x + 5 = 0.(2)

(b)State the minimum possible number of times a different u-shaped quadratic graph could cross the x-axis, and the maximum possible number.(2)

(Total for Question 11 is 4 marks)

A quadratic graph has the equation y = (x - 1)(x - 5).

(a)Write down the coordinates of the two points where the graph crosses the x-axis.(2)

(b)Find the coordinates of the point where the graph crosses the y-axis.(2)

(c)Using the symmetry of the graph, find the coordinates of the turning point.(2)

(Total for Question 12 is 6 marks)

The grid shows an accurately drawn graph of y = x² - 2x - 1 for -2 ≤ x ≤ 4. By drawing a suitable straight line on the grid, use the graph to find estimates for the solutions of x² - 2x - 1 = x - 1.

(Total for Question 13 is 3 marks)

A quadratic graph has the equation y = (x - 2)² - 3.

(a)Write down the coordinates of the turning point of the graph.(1)

(b)State the equation of the line of symmetry of the graph.(1)

(c)Show that y = (x - 2)² - 3 can be written in the form y = x² - 4x + 1.(2)

(d)Hence write down the coordinates of the y-intercept of the graph.(1)

(Total for Question 14 is 5 marks)

The graph of y = x² - 4x - 2 is to be drawn for -2 ≤ x ≤ 6.

(a)Complete the table of values for y = x² - 4x - 2.
x : -2 , -1 , 0 , 1 , 2 , 3 , 4 , 5 , 6
y : 10 , _ , -2 , _ , -6 , _ , -2 , _ , 10(2)

(b)Draw the graph of y = x² - 4x - 2 for -2 ≤ x ≤ 6.(2)

(c)By drawing a suitable straight line on the grid, use your graph to find estimates for the solutions of x² - 4x - 2 = 2x - 5.(3)

(Total for Question 15 is 7 marks)

The graphs of y = x² - 3x + 1 and y = 2x - 4 intersect at two points.

(a)Show that the x-coordinates of the points of intersection satisfy x² - 5x + 5 = 0.(2)

(b)Solve x² - 5x + 5 = 0, giving your answers to 2 decimal places.(3)

(c)Hence find the coordinates of the two points of intersection, giving your answers to 2 decimal places.(2)

(Total for Question 16 is 7 marks)

Mark scheme · 5.9 Drawing Quadratic Graphs

Question 1

(a) B1 at least 5 of the 7 y-values correct
(a) B1 all 7 y-values correct
(a) Answer: 7, 2, -1, -2, -1, 2, 7
(b) B1 all points plotted correctly ft from table (± 1 small square)
(b) B1 smooth u-shaped curve drawn through all points, not a series of straight lines
(b) Answer: Smooth u-shaped parabola through the 7 points, minimum point at (0,-2)
(c) B1 answer in the range 0.1 to 0.4, ft from candidate's graph
(c) Answer: y = awrt 0.25 (accept 0.1 to 0.4)
(d) B1 x = awrt 2.6 (range 2.5 to 2.7), ft from candidate's graph
(d) B1 x = awrt -2.6 (range -2.7 to -2.5), ft from candidate's graph
(d) Answer: x = awrt 2.6 or x = awrt -2.6

Question 2

(a) B1 at least 5 of the 7 y-values correct
(a) B1 all 7 y-values correct
(a) Answer: -5, 0, 3, 4, 3, 0, -5
(b) B1 all points plotted correctly ft from table
(b) B1 smooth n-shaped curve drawn through all points
(b) Answer: Smooth n-shaped parabola, maximum point at (0,4)
(c) B1 (-2, 0)
(c) B1 (2, 0)
(c) Answer: (-2, 0) and (2, 0)
(d) B1 (0, 4)
(d) Answer: (0, 4)

Question 3

(a) B1 at least 4 of the 6 A-values correct
(a) B1 all 6 A-values correct
(a) Answer: 4, 10, 18, 28, 40, 54
(b) B1 all points plotted correctly ft from table
(b) B1 smooth increasing curve drawn through all points
(b) Answer: Smooth increasing curve (part of a parabola) through the 6 points
(c) B1 reads a horizontal line from A = 40 to the curve, ft from candidate's graph
(c) B1 x = 5 (accept 4.8 to 5.2)
(c) Answer: x = 5 (metres)

Question 4

(a) B1 A
(a) Answer: A
(b) B1 correct substitution check shown for at least one value, e.g. at x=-3, (-3)²-3(-3)=9+9=18 oe
(b) Answer: At x = -3: (-3)² - 3(-3) = 9 + 9 = 18, which only matches table A

Question 5

(i) B1 u-shaped
(i) Answer: u-shaped
(ii) B1 n-shaped
(ii) Answer: n-shaped
(iii) B1 u-shaped
(iii) Answer: u-shaped
(iv) B1 n-shaped
(iv) Answer: n-shaped

Question 6

(a) B1 at least 5 of the 7 y-values correct
(a) B1 all 7 y-values correct
(a) Answer: 5, 0, -3, -4, -3, 0, 5
(b) B1 all points plotted correctly ft from table
(b) B1 smooth u-shaped curve drawn through all points
(b) Answer: Smooth u-shaped parabola, minimum point at (-1,-4)
(c) B1 x = -3
(c) B1 x = 1
(c) Answer: x = -3 and x = 1
(d) B1 (-1, -4)
(d) Answer: (-1, -4)

Question 7

(a) B1 h = 18.75 at t = 1.5
(a) B1 h = 18.75 at t = 2.5
(a) B1 h = 8.75 at t = 3.5
(a) Answer: t=1.5: 18.75, t=2.5: 18.75, t=3.5: 8.75
(b) B1 all 9 points plotted correctly ft from table
(b) B1 smooth n-shaped curve drawn through all points
(b) Answer: Smooth n-shaped curve, maximum point at (2, 20)
(c) B1 20 metres, ft from candidate's graph
(c) Answer: 20 metres (at t = 2 seconds)
(d) B1 t = awrt 0.7 (range 0.6 to 0.8), ft from candidate's graph
(d) B1 t = awrt 3.3 (range 3.2 to 3.4), ft from candidate's graph
(d) Answer: t = awrt 0.7 seconds and t = awrt 3.3 seconds

Question 8

(a) B1 x = -2
(a) B1 x = 3
(a) Answer: x = -2 and x = 3
(b) M1 line y = 4 drawn on the grid
(b) A1 x = awrt 3.7 (range 3.6 to 3.8), ft
(b) A1 x = awrt -2.7 (range -2.8 to -2.6), ft
(b) Answer: x = awrt 3.7 and x = awrt -2.7

Question 9

(a) B1 n-shaped
(a) B1 reason: coefficient of x² is negative oe
(a) Answer: n-shaped, because the coefficient of x² is negative
(b) B1 (0, 4)
(b) Answer: (0, 4)
(c) B1 (0, 4)
(c) Answer: (0, 4)
(d) B1 B
(d) Answer: B

Question 10

(a) B1 y = -4.25 at x = -0.5
(a) B1 y = -4.25 at x = 0.5
(a) Answer: -4.25 and -4.25
(b) B1 all 7 points plotted correctly ft from table
(b) B1 smooth u-shaped curve drawn through all points
(b) Answer: Smooth u-shaped curve, minimum point at (0,-5)
(c) B1 x = awrt 1.3 (range 1.2 to 1.4), ft
(c) B1 x = awrt -1.3 (range -1.4 to -1.2), ft
(c) Answer: x = awrt 1.3 and x = awrt -1.3

Question 11

(a) B1 there are no real solutions/roots
(a) B1 correct reason: the curve never crosses (or touches) the x-axis oe
(a) Answer: There are no real solutions, because the curve never crosses (or touches) the x-axis
(b) B1 minimum = 0
(b) B1 maximum = 2
(b) Answer: Minimum 0, maximum 2

Question 12

(a) B1 (1, 0)
(a) B1 (5, 0)
(a) Answer: (1, 0) and (5, 0)
(b) M1 substitutes x = 0 into y = (x-1)(x-5)
(b) A1 (0, 5) cao
(b) Answer: (0, 5)
(c) M1 x = 3 (midpoint of the roots x=1 and x=5)
(c) A1 (3, -4) cao
(c) Answer: (3, -4)

Question 13

M1 line y = x - 1 identified and drawn on the grid
A1 x = 0 (or awrt 0), ft
A1 x = 3 (or awrt 3), ft
Answer: x = 0 and x = 3

Question 14

(a) B1 (2, -3)
(a) Answer: (2, -3)
(b) B1 x = 2
(b) Answer: x = 2
(c) M1 expands (x-2)² to x² - 4x + 4
(c) A1 correct simplification to x² - 4x + 1 cso
(c) Answer: x² - 4x + 1 (shown)
(d) B1 (0, 1) ft from part (c)
(d) Answer: (0, 1)

Question 15

(a) B1 at least 2 of the 4 missing y-values correct
(a) B1 all 4 missing y-values correct
(a) Answer: x=-1: 3, x=1: -5, x=3: -5, x=5: 3
(b) B1 all 9 points plotted correctly ft from table
(b) B1 smooth u-shaped curve drawn through all points
(b) Answer: Smooth u-shaped curve, minimum point at (2,-6)
(c) M1 line y = 2x - 5 identified and drawn on the grid
(c) A1 x = awrt 0.6 (range 0.5 to 0.7), ft
(c) A1 x = awrt 5.4 (range 5.3 to 5.5), ft
(c) Answer: x = awrt 0.6 and x = awrt 5.4

Question 16

(a) M1 sets x²-3x+1 = 2x-4 and begins to rearrange
(a) A1 correct simplification to x² - 5x + 5 = 0 cso
(a) Answer: x² - 5x + 5 = 0 (shown)
(b) M1 correct substitution into the quadratic formula, e.g. x=(5±√25-20)/2
(b) A1 x = 3.62 (awrt)
(b) A1 x = 1.38 (awrt)
(b) Answer: x = 3.62 or x = 1.38 (2 d.p.)
(c) M1 substitutes both x-values into y = 2x - 4 (or the other equation)
(c) A1 (3.62, 3.24) and (1.38, -1.24), ft
(c) Answer: (3.62, 3.24) and (1.38, -1.24)

Drawing Quadratic Graphs - Worksheets, Questions and Revision