Advanced problem solving with non-linear graphs | Maths

Starter quiz

__________ is a line that a curve approaches but never touches.

A tangent

A linear graph

An asymptote ✓

An axis
Laura plots the graph of $y=x^3$ . When the $x$ coordinate is $-4$ , the $y$ coordinate is ______.

'-64' ✓
Andeep plots the reciprocal graph $y={1\over{x}}$ . Which of these will be coordinate pairs?

$(3,{1\over3})$ ✓

$(0,0)$

$(-2,2)$

$(-2,-2)$

$({1\over4},4)$ ✓
Jun graphs the circle $x^2+y^2=16$ . When the $y$ coordinate is $3$ , the $x$ coordinate is ______.

$5$

$4$

$\sqrt7$ ✓

$-\sqrt7$ ✓

$5$
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 $x^2+y^2=25$ ?

$25$

$25^2$

$5$ ✓

$-5$
A quarter circle with radius 7 overlaps a square of length 7. Which of these calculations will give the shaded area?

$7^2-{{1\over{4}}\times{\pi}\times7^2}$ ✓

$7^2-{{1\over{4}}\times{\pi}\times14}$

$49-49{\pi}$

$7^2-{{1\over{4}}\times{\pi}\times14^2}$

Exit quiz

This is an example of __________ graph.

a cubic

an exponential

a quadratic

a reciprocal ✓
This is an example of __________ graph.

a cubic

an exponential ✓

a quadratic

a reciprocal
$How can you tell that this is not the graph of <Math>y={5\over{x}}</Math>?$

How can you tell that this is not the graph of $y={5\over{x}}$ ?

It is not a reciprocal graph.

$y={5\over{x}}$ would not have the axes as asymptotes.

For $y={5\over{x}}$ when $x$ is positive $y$ would be positive. ✓

For $y={5\over{x}}$ when $x$ is negative $y$ would be negative. ✓

This might be the graph $y={5\over{x}}$
Which of these could be the equation of this graph?

$y=x^3-3$

$y={-3\over{x}}$

$y=3-x^3$

$y={-3\over{x}}-3$

$y=-x^3-3$ ✓
Which of these could be the equation of this graph?

$y={4\over{x}}-2$ ✓

$y=2-{4\over{x}}$

$y=2-4^x$

$y=-4-2^x$

$y={4\over{x}}$
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?

$\pi\times7^2$

${1\over2}\times\pi\times7^2$

${1\over4}\times\pi\times49^2$

${1\over4}\times\pi\times7^2$

${1\over8}\times\pi\times7^2$ ✓

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Lesson Details

Key learning points

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

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.

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.