New
New
Year 10
Higher

Key features of a cubic graph

I can identify the key features of a cubic graph.

New
New
Year 10
Higher

Key features of a cubic graph

I can identify the key features of a cubic graph.

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Lesson details

Key learning points

  1. A cubic graph has a distinct shape.
  2. The roots of a cubic graph are where the graph intersects with the x-axis.
  3. The turning points are the local maximum and local minimum points of the graph.

Keywords

  • Cubic - A cubic is an equation, graph, or sequence whereby the highest exponent of the variable is 3

  • Roots - When drawing the graph of an equation, the roots of the equation are where its graph intercepts the x-axis (where y = 0).

  • Turning point - The turning point of a graph is a point on the curve where, as x increases, the y values change from decreasing to increasing or vice versa.

Common misconception

$$y=x^3$$ has one root so all cubic graphs have one root.

Make sure pupils see a wide variety of cubic graphs; ones with one root, two roots and three roots and link every one back to its equation and that the highest exponent of the variable is $$3$$ so they are all cubic graphs.

Have Desmos.com on a screen visible to pupils during the lesson and get them to suggest cubic equations to graph. Get pupils to pick out the key features of their equations.
Teacher tip

Licence

This content is © Oak National Academy Limited (2024), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

Lesson video

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6 Questions

Q1.
A cubic is an equation, graph, or sequence where the highest __________ of the variable is 3.
base
coefficient
Correct answer: exponent
factor
multiple
Q2.
How many solutions can a quadratic equation have?
1
1 or 2
2
0 or 1
Correct answer: 0, 1 or 2
Q3.
What is $$y$$-intercept of this quadratic graph?
An image in a quiz
8
$$y$$ = 8
Correct answer: (0, 8)
(4, 0)
(2, 0)
Q4.
What are the roots of this quadratic equation?
An image in a quiz
(2, 0)
(4, 0)
Correct answer: $$x$$ = 2
Correct answer: $$x$$ = 4
$$y$$ = 8
Q5.
The point (3, -1) is the point of this quadratic curve.
An image in a quiz
Correct Answer: minimum, turning
Q6.
Factorise $$x^2-6x$$.
$$(x+1)(x-6)$$
$$(x-1)(x-6)$$
Correct answer: $$x(x-6)$$
$$(x-1)(x+6)$$
$$(x+1)(x+6)$$

6 Questions

Q1.
The point on the graph of a curve where, as $$x$$ increases, the $$y$$ values change from decreasing to increasing (or vice versa) is called the __________.
root
Correct answer: turning point
$$x$$-intercept
$$y$$-intercept
Q2.
How many roots can the graph of a cubic equation have?
1
3
1 or 3
Correct answer: 1, 2 or 3
0, 1, 2 or 3
Q3.
This is the graph of $$y=x^{3}-9x^{2}+24x-16$$. Which $$y$$ values make this statement true? $$y=x^{3}-9x^{2}+24x-16$$ has two solutions.
An image in a quiz
5
Correct answer: 4
2
Correct answer: 0
-3
Q4.
This is the equation $$y=x^{3}-9x^{2}+24x-16$$. The root $$x=4$$ is a __________.
An image in a quiz
Correct answer: repeated root
minimum value
Correct answer: turning point
Correct answer: local minimum
local maximum
Q5.
The equation $$y=x^3-21x+20$$ factorises to $$y=(x-1)(x-4)(x+5)$$. What are the roots of the equation?
Correct answer: $$x=1$$
Correct answer: $$x=4$$
$$x=-1$$
$$x=-4$$
Correct answer: $$x=-5$$
Q6.
$$x=-5$$ and $$x=0$$ are two roots of $$y=x^3+2x^2-15x$$. The third root is $$x=$$ .
Correct Answer: 3, x=3