Graphs of Functions (Edexcel International AS Maths: Pure 1): Exam Questions

Sketch the graph of $y=(x+3{)}^3$ labelling any points where the graph intersects the coordinate axes.

Sketch the graph of $y=\frac{1}{x^2}$ and write down the equations of any asymptotes.

On the axes below sketch the graphs of both $y=3x-2$ and $y=x+4$.

Using your graph, or otherwise, find the solution to the simultaneous equations

$3x-y=2$ and

$x-y=-4$.

$y$ is inversely proportional to $x$.  When $x=3,y=12$.  Find the constant of proportionality and sketch the graph of $y$ against $x$.

Sketch the graph of $y=2x^2(x+3)$ labelling any points where the graph intersects the coordinate axes.

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

Use your graph to determine the number of solutions to the equation

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

A machine computes a calculation in time, $t$ seconds, that is proportional to the number of processes, $p$, involved.  For a calculation involving 10 processes the machine takes 0.01 seconds.

Show that the constant of proportionality is 0.001.

Hence write down an equation linking the number of processes $\left(p\right)$ to the time taken $\left(t\right)$.

Find the time it takes for the machine to compute a calculation involving 200 processes.

How many processes are involved for a calculation taking 2.3 seconds?

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

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

State which graph the following points must lie on and find the value of $a$ in each case.

(i) $(-4,-5)$ 

(ii) $(-0.02,250)$

Solve the equation $x^3-x^2-2x+4=4x+4$.

Write down the $x$-coordinates of the points of intersection between the graphs of $y=x^3-x^2-2x+4$ and $y=4x+4$.

Find the $y$-coordinates of the points of intersection between the graphs of $y=x^3-x^2-2x+4$ and $y=4x+4$.

On the same diagram, sketch the graphs of $y=x^3-x^2-2x+4$ and $y=4x+4$.

Sketch the graph of $y=\frac{-2}{x^2}$ and write down the equations of any asymptotes.

On the axes below sketch the graphs of both $y=x^2+2x-3$ and $y=x-1$.

Using your graph, or otherwise, find the solutions to the simultaneous equations

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

$x-y=1$

$y$ is inversely proportional to $x$.  When $x=2,y=10$.  Find the constant of proportionality and sketch the graph of $y$ against $x$.

Sketch the graph of $y=3x^3-2x^2-x$ labelling any points where the graph intersects the coordinate axes.

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

Use your graph to determine the number of solutions to the equation

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

A machine computes a calculation in time, $t$ seconds, that is proportional to the square of the number of processes, $p$, involved.  For a calculation involving 8 processes the machine takes 0.032 seconds.

Find an equation linking the number of processes, ($p$) to the time taken, ($t$) seconds.

How many processes are involved for a calculation taking 0.2 seconds?

Find the time it takes for the machine to compute a calculation involving 30 processes.

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

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

Given that $m$ is a negative real number, state with a reason, which graph passes through the point $\left(m,m^4\right)$.

On the same diagram, sketch the graphs of $y=\frac{1}{x^2}$ and $y=\frac{-3}{x^2}$.

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

On the axes below sketch the graphs of both $y=(x-1{)}^2$ and $y=2-x^2-x$.

Using your graph, or otherwise, find the solutions to the equation

$x^2-2x+1=2-x^2-x$.

$y$ is inversely proportional to the square of $x$.  When $x=4,y=8$.  Find the constant of proportionality and sketch the graph of $y$ against $x$.

A machine computes a calculation in time, $t$ seconds, that is proportional to the cube root of the number of processes, $p$, involved.  For a calculation involving 8 processes the computer takes $6.4\times {10}^{-4}$ seconds.

How many processes are involved for a calculation taking $1.28\times {10}^{-3}$ seconds?

Find the time it takes for the machine to compute a calculation involving 250 processes.

On separate diagrams, sketch the graphs of $y=\frac{a}{x^2}$, where $a>0$ and $y=\frac{a}{x^2}$, where $a<0$.

One of the graphs passes through the point with coordinates $\left(m,m^6\right)$.

Write $a$ in terms of $m$ and, justifying your answer, state which graph this point must lie on.