Cubic, reciprocal, exponential and circle graphs

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Cubic graphs

A cubic has equation y = ax³ + bx² + cx + d with a ≠ 0. The simplest, y = x³, passes through the origin and has rotational symmetry.
A cubic with positive x³ rises from bottom-left to top-right; with negative x³ it falls. It can cross the x-axis up to three times.

Reciprocal graphs

For k ≠ 0, y = k/x is a hyperbola with two separate branches. It is undefined at x = 0.
It never touches the axes: the axes are asymptotes. x = 0 is undefined.
The graph is symmetrical about the lines y = x and y = -x.

Cubic and reciprocal curves

Exponential graphs

y = a^x (with a > 0, a \neq 1) grows or decays rapidly. y = 2^x passes through (0, 1) because 2⁰ = 1.
For a > 1 the graph rises ever more steeply; for 0 < a < 1 (e.g. y = (\frac{1}{2})^x) it decays towards the x-axis, which is an asymptote.
Never negative: y > 0 for all x.

Exponential growth

Circle graphs

A circle with centre the origin and radius r has equation x² + y² = r². For x² + y² = 25, the radius is 5.
To check if a point lies on the circle, substitute it: (3, 4) gives 9 + 16 = 25, so it does.
The tangent to a circle is perpendicular to the radius at the point of contact.

x² + y² = 25

Recognising graphs

Link the shape to the equation: line, parabola, cubic S-shape, hyperbola, exponential curve, circle.

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Practice questions

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1.What is the shape of the graph of y = \frac{1}{x} called?
Easy
- Aparabola
- Bhyperbola
- Cellipse
- Dcubic
2.What is y when x = 0 on y = 2^x?
Easy
- A0
- B1
- C2
- Dundefined
3.What is y when x = 3 on y = 2^x?
Easy
- A6
- B8
- C9
- D5
4.What is y when x = 2 on y = x³?
Easy
- A6
- B8
- C9
- D5
5.What is y when x = -2 on y = x³?
Easy
- A8
- B-8
- C6
- D-6
6.What is y when x = 4 on y = \frac{1}{x}?
Easy
- A4
- B0.25
- C0.4
- D-4
7.What is y when x = -2 on y = \frac{6}{x}?
Easy
- A3
- B-3
- C12
- D-12
8.Which value of x can't be used in y = \frac{1}{x}?
Easy
- A1
- B0
- C-1
- D10

More questions

9.What does the graph of y = 2^x approach as x becomes very negative?
Easy
- Ay = 1
- By = 0
- Cy = -1
- Dy becomes large
10.What is the radius of x² + y² = 49?
Easy
- A7
- B49
- C24.5
- D14
11.What is the radius of x² + y² = 10?
Easy
- A5
- B\√{10}
- C10
- D100
12.Which point lies on x² + y² = 25?
Medium
- A(3, 4)
- B(4, 4)
- C(5, 5)
- D(2, 3)
13.Which point lies on x² + y² = 13?
Medium
- A(2, 3)
- B(3, 3)
- C(1, 4)
- D(3, 4)
14.Which equation gives a circle of radius 6 centred at the origin?
Medium
- Ax² + y² = 6
- Bx² + y² = 12
- Cx² + y² = 36
- Dx + y = 6
15.Which equation matches a graph that passes through (0, 1) and rises steeply?
Medium
- Ay = 3^x
- By = x³
- Cy = \frac{1}{x}
- Dy = x²
16.Which graph is symmetrical about the origin (rotational symmetry) and S-shaped?
Medium
- Ay = x²
- By = x³
- Cy = 2^x
- Dy = |x|
17.Which graph has the x-axis as an asymptote?
Medium
- Ay = x³
- By = 2^x
- Cy = x²
- Dy = 2x
18.What is y when x = -1 on y = 2^x?
Medium
- A-2
- B0.5
- C-0.5
- D2
19.The curve y = 10 \times 2^x passes through (0, k). What is k?
Medium
- A1
- B10
- C20
- D0
20.A tangent touches the circle x² + y² = 25 at (3, 4). What is the gradient of the radius to that point?
Medium
- A\frac{4}{3}
- B\frac{3}{4}
- C-\frac{3}{4}
- D-\frac{4}{3}
21.The tangent at (3, 4) to x² + y² = 25 has gradient:
Medium
- A\frac{4}{3}
- B\frac{3}{4}
- C-\frac{3}{4}
- D-\frac{4}{3}
22.The graph y = \frac{1}{x} lies in which quadrants?
Easy
- A1st and 2nd
- B1st and 3rd
- C2nd and 4th
- D3rd and 4th
23.Which table gives y = x³ for x = -1, 0, 1?
Easy
- A-1, 0, 1
- B1, 0, 1
- C-3, 0, 3
- D1, 0, -1
24.For y = \frac{1}{x}, what happens to y as x increases from 1 to 100?
Medium
- Ay increases
- By decreases towards 0
- Cy stays the same
- Dy becomes negative
25.Which has no x-intercept?
Medium
- Ay = x³
- By = 2^x
- Cy = x²
- Dy = x - 1
26.What is the centre of the circle x² + y² = 9?
Easy
- A(3, 3)
- B(0, 0)
- C(9, 0)
- D(0, 9)
27.y = x³ passes through the origin.
Easy
True or false?
28.The graph of y = 2^x can be negative.
Easy
True or false?
29.x² + y² = 25 is the equation of a circle of radius 5 centred at the origin.
Easy
True or false?
30.The graph of y = \frac{1}{x} crosses the y-axis.
Easy
True or false?
31.The point (4, 4) lies on x² + y² = 25.
Easy
True or false?
32.The tangent to a circle is perpendicular to the radius at the point of contact.
Medium
True or false?
33.Which graphs pass through (0, 1)?
Medium
- Ay = 2^x
- By = x² + 1
- Cy = \frac{1}{x}
- Dy = x³
- Ey = 5^x
34.Which points lie on x² + y² = 25?
Medium
- A(5, 0)
- B(0, -5)
- C(3, 4)
- D(4, 4)
- E(2, 5)
35.Describe the graph of y = 2^x.
Medium

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