Sketching Curves (Copy)
Question 1
Sketch the straight-line graph of y = 2x + 3.
State the gradient and y-intercept.
Question 2
Sketch the quadratic y = x² – 4x + 3.
Mark the roots and turning point.
Question 3
Sketch the cubic y = x³.
Describe its symmetry.
Question 4
Sketch the reciprocal graph y = 1/x.
State the equations of the asymptotes.
Question 5
Sketch the exponential graph y = 2Ë£.
State the asymptote.
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
Sketch y = –3x + 1.
Give its gradient and intercept.
Question 7
Sketch y = (x – 2)² – 1.
Identify the minimum point.
Question 8
Sketch y = –x² + 4.
State the maximum value.
Question 9
Sketch y = x³ – 3x.
Mark the roots and turning points.
Question 10
Sketch y = 1/(x – 2).
State the vertical and horizontal asymptotes.
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
Sketch y = 3Ë£ + 1.
State the asymptote and y-intercept.
Question 12
Sketch y = –2x + 5.
Find its gradient and y-intercept.
Question 13
Sketch y = x² – 9.
State the roots and axis of symmetry.
Question 14
Sketch y = (x + 2)(x – 4).
Mark the roots and turning point.
Question 15
Sketch y = x³ – x² – x + 1.
Mark approximate roots and describe its shape.
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
Sketch y = 1/x².
Explain why it is always positive.
Question 17
Sketch y = –1/x.
State the asymptotes and quadrants.
Question 18
Sketch y = 5ˣ – 2.
State the asymptote and y-intercept.
Question 19
Sketch y = –x³.
Explain the symmetry.
Question 20
Sketch y = 2x – 4.
State the gradient and y-intercept.
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
Sketch y = x² + 6x + 8.
Find roots and turning point.
Question 22
Sketch y = –(x – 1)² + 5.
Identify maximum point.
Question 23
Sketch y = x³ – 2x² – x + 2.
Indicate approximate turning points.
Question 24
Sketch y = 1/(x + 1).
State asymptotes.
Question 25
Sketch y = 4Ë£.
Show asymptote and growth pattern.
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
Sketch y = –x + 7.
Give gradient and intercept.
Question 27
Sketch y = (x – 3)².
Identify minimum point and axis of symmetry.
Question 28
Sketch y = x² – 2x – 8.
Find the roots graphically.
Question 29
Sketch y = –x³ + 2x.
State turning points and intercepts.
Question 30
Sketch y = 2Ë£ + 4.
Give asymptote and intercept.
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 = 2x + 3
- Gradient = 2 (steep upward slope).
- y-intercept = (0,3). Straight line through (0,3).
2.
y = x² – 4x + 3
- Factor: (x – 1)(x – 3). Roots: x = 1, 3.
- Axis of symmetry: x = 2.
- Vertex: (2, –1).
3.
y = x³
- Passes through (0,0).
- Symmetry about the origin (rotational).
4.
y = 1/x
- Two branches in quadrants I and III.
- Vertical asymptote: x = 0. Horizontal asymptote: y = 0.
5.
y = 2Ë£
- Always positive, passes through (0,1).
- Horizontal asymptote: y = 0.
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 = –3x + 1
- Gradient = –3 (steep downward).
- y-intercept = (0,1).
7.
y = (x – 2)² – 1
- Minimum at (2, –1).
- Axis of symmetry: x = 2.
8.
y = –x² + 4
- Inverted parabola.
- Vertex (0,4). Maximum value = 4.
9.
y = x³ – 3x
- Factor: x(x – √3)(x + √3). Roots: x = 0, ±√3 ≈ ±1.73.
- Turning points near (–1,2) and (1, –2).
10.
y = 1/(x – 2)
- Vertical asymptote: x = 2.
- Horizontal asymptote: y = 0.
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 = 3Ë£ + 1
- y-intercept = (0,2).
- Horizontal asymptote: y = 1.
12.
y = –2x + 5
- Gradient = –2.
- y-intercept = (0,5).
13.
y = x² – 9
- Roots: x = ±3.
- Axis of symmetry: x = 0.
- Vertex: (0, –9).
14.
y = (x + 2)(x – 4)
- Roots: x = –2, 4.
- Axis: x = 1.
- Vertex at (1, –9).
15.
y = x³ – x² – x + 1
- Approx roots: by factorisation (x – 1)(x² – 1) = (x – 1)(x – 1)(x + 1).
- Roots: –1, 1 (double).
- Shape: cubic with flattening at x = 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
16.
y = 1/x²
- Always positive.
- Vertical asymptote: x = 0.
- Horizontal asymptote: y = 0.
17.
y = –1/x
- Branches in quadrants II and IV.
- Asymptotes: x = 0, y = 0.
18.
y = 5ˣ – 2
- y-intercept: (0, –1).
- Horizontal asymptote: y = –2.
19.
y = –x³
- Symmetry about origin.
- Falls steeply, passes through (0,0).
20.
y = 2x – 4
- Gradient = 2.
- y-intercept: (0, –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 = x² + 6x + 8 = (x + 2)(x + 4)
- Roots: –2, –4.
- Axis: x = –3.
- Vertex: (–3, –1).
22.
y = –(x – 1)² + 5
- Vertex (1,5).
- Maximum value = 5.
23.
y = x³ – 2x² – x + 2
- Roots approx at x = –1, 1, 2.
- Turning points near (0.5, positive small) and (1.5, negative small).
24.
y = 1/(x + 1)
- Vertical asymptote: x = –1.
- Horizontal asymptote: y = 0.
25.
y = 4Ë£
- y-intercept: (0,1).
- Horizontal asymptote: y = 0.
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 = –x + 7
- Gradient = –1.
- y-intercept: (0,7).
27.
y = (x – 3)²
- Minimum point: (3,0).
- Axis of symmetry: x = 3.
28.
y = x² – 2x – 8 = (x – 4)(x + 2)
- Roots: x = –2, 4.
- Vertex: (1, –9).
29.
y = –x³ + 2x
- Roots: x = 0, ±√2 ≈ ±1.41.
- Turning points near (–0.82, 1.09) and (0.82, –1.09).
30.
y = 2Ë£ + 4
- y-intercept: (0,5).
- Horizontal asymptote: y = 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