Graphs of Functions (AQA A Level Maths: Pure): Exam Questions

Sketch the curve with equation

$y=-\frac{2}{x^2}$

stating the equations of any asymptotes.

On the same axes below, plot the curve $y=x^2+2x-3$ and the line $y=x-1$.

Hence use a graphical method to solve the simultaneous equations

$y=(x+3)(x-1)$

$x-y=1$

You are given that $y$ is inversely proportional to $x$.

When $x=2$, $y=10$.

(i) Find an equation for $y$ in terms of $x$.

(ii) Sketch the graph of $y$ against $x$, where $x>0$.

The diagram below shows a sketch of the curve $y=\frac{a}{x^2}$, where $a>0$.

Sketch the graph of $y=\frac{a}{x^2}$, where $a<0$.

The graph $y=\frac{a}{x^2}$ passes through the point $\left(m,m^4\right)$, where $m$ is a negative real constant.

Explain whether $a>0$ or $a<0$.

Sketch the curve

$y=3x^3-2x^2-x$

showing the coordinates of any points at which the curve meets the coordinate axes.

On the same coordinate axes, sketch the curves $y=\frac{1}{x^2}$ and $y=-\frac{3}{x^2}$.

Write down the equation(s) of any lines of symmetry and any asymptotes for the two curves in part (a).

Express 

$2x^3+2x^2-12x$

in the form

$ax(x+b)(x+c)$

where $a$, $b$ and $c$ are integers to be found.

Hence sketch the curve with equation

$y=2x^3+2x^2-12x$

showing the coordinates of the points at which the curve cuts the coordinate axes.

Use the factor theorem to show that $(x-2)$ is a factor of the function

$\mathrm{f}\left(x\right)=x^3-2x^2-4x+8$

Hence express $\mathrm{f}\left(x\right)$ as a product of three linear factors.

Sketch the curve $y=\mathrm{f}\left(x\right)$ showing the coordinates of the points at which the curve meets the coordinate axes.

Sketch the curve with equation

$y=(x+3{)}^3$

showing the coordinates of the points at which the curve meets the coordinate axes.

Sketch the curve with equation

$y=\frac{1}{x^n}$

in the case where $n=2$, stating the equations of any asymptotes.

You are given that $y$ is inversely proportional to $x$.

When $x=3$, $y=12$.

(i) Find an equation for $y$ in terms of $x$.

(ii) Sketch the graph of $y$ against $x$, where $x>0$.

Sketch the curve with equation 

$y=2x^2(x+3)$

showing the coordinates of the points at which the curve meets the coordinate axes.

On the same diagram, sketch the curves with equations$y=x(x+2)(x-1)$ and $y=\frac{1}{x}$.

Hence find the number of real solutions to the equation

$x(x+2)(x-1)=\frac{1}{x}$

Show that $(x+2)$ is a factor of $2x^3-3x^2-11x+6$.

Hence express $2x^3-3x^2-11x+6$ in the form

$\left(x+2\right)\left(x+a\right)\left(bx+c\right)$

where $a$, $b$ and $c$ are integers to be found.

Sketch the curve with equation

$y=2x^3-3x^2-11x+6$

showing the coordinates of the points at which the curve meets the coordinate axes.

Given that $(x+1)$ is a factor of $x^3-4x^2+x+6$, sketch the curve

$y=x^3-4x^2+x+6$

showing the coordinates of the points at which the curve meets the coordinate axes.

On the same diagram, sketch the curves with equations $y=x^3-2x^2-8x$ and $y=\frac{1}{x}$.

Show the coordinates of any points where the curves meet the coordinate axes.

Hence find the number of solutions to the equation

$x^3-2x^2-8x=\frac{1}{x}$

By showing that $(x+3)$ is a factor of the function 

$\mathrm{f}\left(x\right)=6x^3+23x^2+11x-12$

fully factorise $\mathrm{f}\left(x\right)$.

Sketch the curve $y=\mathrm{f}\left(x\right)$, showing the coordinates of the points at which the curve meets the coordinate axes.

Find the coordinates of the points of intersection between the curve with equation 

$y=x^3-x^2-4x+4$

and the line with equation

$y=2x+4$

On the same coordinate axes, sketch $y=x^3-x^2-4x+4$ and $y=2x+4$.

Show the coordinates of

• any points where the curve meets the coordinate axes

• any points where the line meets the coordinate axes

• any points of intersection between the curve and the line

You are given that $(x-2)$ and $\left(x+3\right)$are factors of $\mathrm{f}\left(x\right)$, where

$\mathrm{f}\left(x\right)=x^4-9x^3+9x^2+85x-150$

Sketch the curve $y=\mathrm{f}\left(x\right)$ showing the coordinates of the points at which the curve meets the coordinate axes.

The curve $C$ has the equation

$y=3x^3+2x^2-3x+10$

It is known that

• the $x$-coordinate of the minimum point is positive

• the maximum point is in the second quadrant

By finding a factor of $3x^3+2x^2-3x+10$, sketch $C$.

Show the coordinates of any points at which the curve meets the coordinate axes.

Fully factorise

$4x^3+17x^2+20x+4$

Sketch the curve

$y=4x^3+17x^2+20x+4$

showing the coordinates of the points at which the curve meets the coordinate axes.

On the same coordinate axes, sketch the curve 

$4y=x^3-5x^2-12x+36$

and the line

$x+y-6=0$

Show the coordinates of

• any points where the curve meets the coordinate axes

• any points where the line meets the coordinate axes

• any points of intersection between the curve and the line