Graphs of Functions (Copy)
Question 1
Construct a table of values for y = x² – 3x – 4 for –2 ≤ x ≤ 4.
Plot the graph and use it to find the roots of the equation x² – 3x – 4 = 0.
Question 2
Draw the graph of y = x³ – 2x for –3 ≤ x ≤ 3.
From the graph, find the values of x where y = 0.
Question 3
Construct a table and draw the graph of y = 2x + 3/x for x = –4 to –1 and x = 1 to 4.
Explain the behaviour near x = 0.
Question 4
Plot the graph of y = 1/x for –5 ≤ x ≤ –1 and 1 ≤ x ≤ 5.
State the equation of the asymptotes.
Question 5
Plot the curve y = √x + 2 for 0 ≤ x ≤ 9.
Use the graph to solve √x + 2 = 5.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 6
Construct a table of values and plot y = 2x³ – x for –2 ≤ x ≤ 2.
Find the coordinates of the turning point approximately.
Question 7
Draw the graph of y = 2/x² for –4 ≤ x ≤ –1 and 1 ≤ x ≤ 4.
State the asymptotes.
Question 8
Plot the curve y = x³ – 4x + 1 for –3 ≤ x ≤ 3.
Use the graph to solve x³ – 4x + 1 = 0.
Question 9
Draw the graph of y = 4(½)ˣ for 0 ≤ x ≤ 6.
Use the graph to estimate y when x = 2.5.
Question 10
Plot the curve y = 3ˣ for –2 ≤ x ≤ 3.
Use the graph to solve 3Ë£ = 10.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 11
Construct a table of values for y = x³ + x² – 3 for –2 ≤ x ≤ 2.
Plot the graph and solve x³ + x² – 3 = 0.
Question 12
Draw the curve y = –x² + 4 for –3 ≤ x ≤ 3.
Estimate the gradient at x = 1 by drawing a tangent.
Question 13
Construct a table of values and plot y = (x – 1)(x + 3).
Use the graph to find the roots.
Question 14
Draw the graphs of y = x² and y = 2x + 3 on the same axes.
Estimate the solutions to x² = 2x + 3.
Question 15
Plot the curve y = 2ˣ + 1 for –2 ≤ x ≤ 3.
Use the graph to estimate the value of x when y = 6.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 16
Construct a table and draw y = x³ – 3x² + 2.
Use the graph to solve y = 0.
Question 17
Plot y = 1/x + 2 for –5 ≤ x ≤ –1 and 1 ≤ x ≤ 5.
State the vertical and horizontal asymptotes.
Question 18
Draw the curve y = √x for 0 ≤ x ≤ 16.
Use the graph to solve √x = 5.
Question 19
Construct a table and draw y = 3x – x² for –1 ≤ x ≤ 4.
Estimate the maximum value of y.
Question 20
Draw the graphs of y = x² – 2 and y = 2ˣ on the same axes.
Estimate the coordinates of their intersections.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 21
Construct a table and plot y = x³ – 6x for –3 ≤ x ≤ 3.
Find the approximate roots.
Question 22
Draw the curve y = (2/x) + 1 for –5 ≤ x ≤ –1 and 1 ≤ x ≤ 5.
State its asymptotes.
Question 23
Plot the graph y = (x – 2)² + 1 for –2 ≤ x ≤ 5.
Find the minimum value of y from the graph.
Question 24
Construct a table and draw y = 4ˣ – 2 for –1 ≤ x ≤ 3.
Estimate the value of x when y = 20.
Question 25
Draw the graph y = –x³ + 3x for –3 ≤ x ≤ 3.
Use the graph to find the turning points approximately.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 26
Plot the graphs of y = x² and y = 4 – x on the same axes.
Use the graph to solve x² = 4 – x.
Question 27
Construct a table and draw y = 5/x² for –4 ≤ x ≤ –1 and 1 ≤ x ≤ 4.
Identify the asymptotes.
Question 28
Plot the curve y = 2ˣ for –2 ≤ x ≤ 4.
Use the graph to estimate y when x = 2.7.
Question 29
Draw y = x³ – 2x + 1 for –3 ≤ x ≤ 3.
From the graph, estimate the gradient at x = 1.
Question 30
Construct a table and draw y = (x + 1)(x – 2)(x – 3).
Use the graph to estimate all three roots.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
1.
y = x² – 3x – 4
Values: (–2,6), (–1,0), (0,–4), (1,–6), (2,–6), (3,–4), (4,0).
Roots where curve crosses x-axis: about x = –1 and x = 4.
2.
y = x³ – 2x
Table shows symmetry.
Roots when y = 0 → x(x² – 2) = 0 → x = 0, ±√2.
3.
y = 2x + 3/x
Values defined except at x = 0.
As x → 0, function tends to ±∞.
Graph has vertical asymptote at x = 0.
4.
y = 1/x
Hyperbola through quadrants I and III.
Asymptotes: x = 0, y = 0.
5.
y = √x + 2
At y = 5 → √x = 3 → x = 9.
Graph confirms solution around x = 9.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
6.
y = 2x³ – x
Approx table shows turning near x = 0.3.
Curve is cubic, root at x = 0.
7.
y = 2/x²
Always positive, decreases as |x| increases.
Asymptotes: x = 0 (vertical), y = 0 (horizontal).
8.
y = x³ – 4x + 1
Graph crosses x-axis around x ≈ –2.2, 0.25, 1.95.
9.
y = 4(½)ˣ
Decreasing exponential.
At x = 2.5, y ≈ 0.71.
10.
y = 3Ë£
For 3ˣ = 10 → x ≈ log(10)/log(3) ≈ 2.1.
Graph confirms near x = 2.1.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
11.
y = x³ + x² – 3
Graph shows root between x = 1 and 1.2 (approx 1.1).
12.
y = –x² + 4
Parabola inverted.
At x = 1, tangent gradient ≈ –2.
13.
y = (x – 1)(x + 3) = x² + 2x – 3
Roots from graph: x ≈ –3, 1.
14.
y = x² and y = 2x + 3
Intersections where x² = 2x + 3 → x² – 2x – 3 = 0 → (x – 3)(x + 1) = 0 → x = –1, 3.
15.
y = 2Ë£ + 1
When y = 6 → 2ˣ = 5 → x ≈ log(5)/log(2) ≈ 2.32.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
16.
y = x³ – 3x² + 2
Roots found between x = –0.6, 1, and 2.6.
17.
y = 1/x + 2
Vertical asymptote: x = 0.
Horizontal asymptote: y = 2.
18.
y = √x
At y = 5 → √x = 5 → x = 25.
Graph shows solution at 25.
19.
y = 3x – x²
Maximum at vertex (1.5, 2.25).
20.
Graphs y = x² – 2 and y = 2ˣ intersect near (–0.77, –1.4) and (2,4).
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
21.
y = x³ – 6x
Factor: x(x² – 6) → roots at x = 0, ±√6 ≈ ±2.45.
22.
y = (2/x) + 1
Asymptotes: x = 0, y = 1.
23.
y = (x – 2)² + 1
Minimum at (2,1).
24.
y = 4ˣ – 2
When y = 20 → 4ˣ = 22 → x = log(22)/log(4) ≈ 2.23.
25.
y = –x³ + 3x
Turning points where dy/dx = –3x² + 3 = 0 → x = ±1.
Approx values: (–1, –2), (1, 2).
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
26.
y = x² and y = 4 – x
Solve x² = 4 – x → x² + x – 4 = 0.
Roots: (–2.56,6.56) and (1.56,2.44).
27.
y = 5/x²
Vertical asymptote: x = 0.
Horizontal asymptote: y = 0.
28.
y = 2Ë£
At x = 2.7 → y ≈ 6.49.
29.
y = x³ – 2x + 1
Gradient at x = 1: derivative = 3(1)² – 2 = 1.
30.
y = (x + 1)(x – 2)(x – 3)
Roots at x = –1, 2, 3.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course