Horizontal asymptote of y=\frac{x}{x^2+1}?

y=0, because the numerator has lower degree.

Where does a rational curve meet the x-axis?

Where the numerator is zero (and the denominator is not).

Where does it meet the y-axis?

At x=0: substitute into the function.

Why must you not multiply an inequality by the denominator?

It may be negative, which reverses the inequality; multiply by the denominator squared instead.

Critical values for a rational inequality?

Zeros of the numerator and zeros of the denominator.

What does a repeated factor in the numerator do to the graph?

The curve touches the x-axis there instead of crossing it.

How do you find where a curve meets a line?

Equate the equations, multiply out, and solve the resulting quadratic.

Quadratic-theory method for the range of y=\frac{p(x)}{q(x)}?

Form a quadratic in x and require b^2-4ac\ge0.

What do repeated roots at the boundary of the range give?

The stationary points of the curve.

Rational function graphsAQA A-Level Further Maths: Flashcards

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Vertical asymptote of $y=\frac{ax+b}{cx+d}$?

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Vertical asymptote of $y=\frac{ax+b}{cx+d}$ ?: $x=-\frac dc$ , where the denominator is zero.
Horizontal asymptote of $y=\frac{ax+b}{cx+d}$ ?: $y=\frac ac$ .
Horizontal asymptote of $y=\frac{ax^2+bx+c}{dx^2+ex+f}$ ?: $y=\frac ad$ .
Horizontal asymptote of $y=\frac{x}{x^2+1}$ ?: $y=0$ , because the numerator has lower degree.
Where does a rational curve meet the $x$ -axis?: Where the numerator is zero (and the denominator is not).
Where does it meet the $y$ -axis?: At $x=0$ : substitute into the function.
Why must you not multiply an inequality by the denominator?: It may be negative, which reverses the inequality; multiply by the denominator squared instead.
Critical values for a rational inequality?: Zeros of the numerator and zeros of the denominator.
What does a repeated factor in the numerator do to the graph?: The curve touches the $x$ -axis there instead of crossing it.
How do you find where a curve meets a line?: Equate the equations, multiply out, and solve the resulting quadratic.
Quadratic-theory method for the range of $y=\frac{p(x)}{q(x)}$ ?: Form a quadratic in $x$ and require $b^2-4ac\ge0$ .
What do repeated roots at the boundary of the range give?: The stationary points of the curve.
Range of $y=\frac{x}{x^2+1}$ ?: $-\frac12\le y\le\frac12$ .

Exam questions on Rational function graphs

The curve $C$ has equation $y=\dfrac{2x+1}{x-3}$ .
Find the coordinates of the points where $C$ meets the coordinate axes.2 marks
The curve $C$ has equation $y=\dfrac{3x}{x+2}$ .
Determine whether $C$ lies above or below its horizontal asymptote when $x$ is large and positive.2 marks
The curve $C$ has equation $y=\dfrac{x+3}{x-2}$ and the line $L$ has equation $y=5x-9$ .
Show that $C$ and $L$ meet where $x=1$ and where $x=3$ .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).