A cubic has a positive leading coefficient. What happens as x\to-\infty?

y\to-\infty (it starts at the bottom left).

A quartic has a negative leading coefficient. End behaviour?

y\to-\infty at both ends (a \cap shape).

What does a squared factor (x-a)^2 do to the curve at x=a?

The curve touches the x-axis and turns round there.

Quadrants of y=\frac{a}{x^2} for a0?

First and second (always positive).

Why can y=\frac{a}{x} never meet the x-axis?

\frac{a}{x}=0 has no solution when a\neq0.

What is the graph of y\propto x?

A straight line through the origin, with gradient k.

Sketching polynomial and reciprocal graphsAQA A-Level Maths: Flashcards

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What does the leading term decide for a polynomial sketch?

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What does the leading term decide for a polynomial sketch?: The end behaviour: the direction of $y$ as $x\to\infty$ and $x\to-\infty$ .
A cubic has a positive leading coefficient. What happens as $x\to-\infty$ ?: $y\to-\infty$ (it starts at the bottom left).
A quartic has a negative leading coefficient. End behaviour?: $y\to-\infty$ at both ends (a $\cap$ shape).
What does a single factor $(x-a)$ do to the curve at $x=a$ ?: The curve crosses the $x$ -axis.
What does a squared factor $(x-a)^2$ do to the curve at $x=a$ ?: The curve touches the $x$ -axis and turns round there.
How do you find the $y$ -intercept of a curve?: Substitute $x=0$ .
Quadrants of $y=\frac{a}{x}$ for $a>0$ ?: First and third.
Quadrants of $y=\frac{a}{x^2}$ for $a>0$ ?: First and second (always positive).
Asymptotes of $y=\frac{a}{x}$ and $y=\frac{a}{x^2}$ ?: $x=0$ and $y=0$ .
Why can $y=\frac{a}{x}$ never meet the $x$ -axis?: $\frac{a}{x}=0$ has no solution when $a\neq0$ .
$y$ is inversely proportional to $x$ , with $y=3$ at $x=4$ . Find $y$ in terms of $x$ .: $y=\frac{12}{x}$ .
What is the graph of $y\propto x$ ?: A straight line through the origin, with gradient $k$ .
Which is symmetric in the $y$ -axis: $y=\frac1x$ or $y=\frac{1}{x^2}$ ?: $y=\frac{1}{x^2}$ .

Exam questions on Sketching polynomial and reciprocal graphs

The curve $C$ has equation $y=(x+2)(x-1)^2$ .
Write down the coordinates of the point where $C$ touches the $x$ -axis, and describe the behaviour of $y$ as $x\to-\infty$ .2 marks
The curve $D$ has equation $y=\dfrac{4}{x^2}$ .
Write down the equations of the asymptotes of $D$ , and explain why $D$ never meets the $x$ -axis.2 marks
The curve $E$ has equation $y=x^2(3-x)$ .
Find the coordinates of the points where $E$ meets the $x$ -axis and state, with a reason, whether $E$ crosses or touches the $x$ -axis at each of them.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).