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Year 10
Higher

Advanced problem solving with non-linear graphs

I can use my knowledge of non-linear graphs to solve problems.

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New
New
Year 10
Higher

Advanced problem solving with non-linear graphs

I can use my knowledge of non-linear graphs to solve problems.

Download all resources
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Share activities with pupils
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Lesson details

Key learning points

  1. The shape of the graph can be used to identify the form of its equation.
  2. Sketching the graph can help when solving problems.
  3. Problem solving requires you to draw on any useful knowledge.

Keywords

  • Quadratic - A quadratic is an equation, graph, or sequence whereby the highest exponent of the variable is 2

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

  • Exponential - The general form for an exponential equation is y = ab^x where a is the coefficient, b is the base and x is the exponent.

  • Asymptote - An asymptote is a line that a curve approaches but never touches.

Common misconception

A graph drawn on axes with no scale shown means nothing is known about the graph.

Understanding the shape of a parabola, cubic curve, and reciprocal graph and then applying key features like positive/negative coefficients and $$y$$-intercepts enables pupils to infer much about the equation without needing actual coordinate values.

Ask pupils to verbalise their reasoning when matching graphs to equations. "How do you 'know' that is the graph of... ?" is a key question to ask once pupils have matched a pair. Being able to justify with technical vocabulary is a powerful skill in mathematics.
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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Starter quiz

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6 Questions

Q1.
__________ is a line that a curve approaches but never touches.
A tangent
A linear graph
Correct answer: An asymptote
An axis
Q2.
Laura plots the graph of $$y=x^3$$. When the $$x$$ coordinate is $$-4$$, the $$y$$ coordinate is .
Correct Answer: -64, y=-64
Q3.
Andeep plots the reciprocal graph $$y={1\over{x}}$$. Which of these will be coordinate pairs?
Correct answer: $$(3,{1\over3})$$
$$(0,0)$$
$$(-2,2)$$
$$(-2,-2)$$
Correct answer: $$({1\over4},4)$$
Q4.
Jun graphs the circle $$x^2+y^2=16$$. When the $$y$$ coordinate is $$3$$, the $$x$$ coordinate is .
$$5$$
$$4$$
Correct answer: $$\sqrt7$$
Correct answer: $$-\sqrt7$$
$$5$$
Q5.
The general form of the equation of a circle with centre origin is $$x^2+y^2=r^2$$. What is the radius of the circle graphed by the equation $$x^2+y^2=25$$?
$$25$$
$$25^2$$
Correct answer: $$5$$
$$-5$$
Q6.
A quarter circle with radius 7 overlaps a square of length 7. Which of these calculations will give the shaded area?
An image in a quiz
Correct answer: $$7^2-{{1\over{4}}\times{\pi}\times7^2}$$
$$7^2-{{1\over{4}}\times{\pi}\times14}$$
$$49-49{\pi}$$
$$7^2-{{1\over{4}}\times{\pi}\times14^2}$$

Exit quiz

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6 Questions

Q1.
This is an example of __________ graph.
An image in a quiz
a cubic
an exponential
a quadratic
Correct answer: a reciprocal
Q2.
This is an example of __________ graph.
An image in a quiz
a cubic
Correct answer: an exponential
a quadratic
a reciprocal
Q3.
How can you tell that this is not the graph of $$y={5\over{x}}$$?
An image in a quiz
It is not a reciprocal graph.
$$y={5\over{x}}$$ would not have the axes as asymptotes.
Correct answer: For $$y={5\over{x}}$$ when $$x$$ is positive $$y$$ would be positive.
Correct answer: For $$y={5\over{x}}$$ when $$x$$ is negative $$y$$ would be negative.
This might be the graph $$y={5\over{x}}$$
Q4.
Which of these could be the equation of this graph?
An image in a quiz
$$y=x^3-3$$
$$y={-3\over{x}}$$
$$y=3-x^3$$
$$y={-3\over{x}}-3$$
Correct answer: $$y=-x^3-3$$
Q5.
Which of these could be the equation of this graph?
An image in a quiz
Correct answer: $$y={4\over{x}}-2$$
$$y=2-{4\over{x}}$$
$$y=2-4^x$$
$$y=-4-2^x$$
$$y={4\over{x}}$$
Q6.
The circle $$x^2+y^2=49$$ and the line $$x+y=0$$ are drawn on the same axes. What is the area of the region bound by the circumference of the circle, the line $$x+y=0$$, and the $$y$$ axis?
$$\pi\times7^2$$
$${1\over2}\times\pi\times7^2$$
$${1\over4}\times\pi\times49^2$$
$${1\over4}\times\pi\times7^2$$
Correct answer: $${1\over8}\times\pi\times7^2$$