Graphs of Functions (Cambridge (CIE) A Level Maths: Pure 1): Exam Questions

Express $2x^3+2x^2-12x$ in the form $ax(x+b)(x+c)$, where $a$, $b$ and $c$ are integers.

Hence sketch the graph of $y=2x^3+2x^2-12x$, giving the coordinates of the points where the graph meets the coordinate axes.

Sketch the graph of $y=(x+3{)}^3$, giving the coordinates of the points where the graph meets 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 $y=3x-2$ and $y=x+4$.

Using your graphs, or otherwise, solve 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 $y=2x^2(x+3)$, giving the coordinates of the points where the graph meets the coordinate axes.

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

Use your diagram to determine the number of solutions of the equation $x(x+2)(x-1)=\frac{1}{x}$.

A machine completes a calculation in a 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 $p$ and $t$.

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

How many processes are involved in a calculation taking $2.3$ seconds?

The diagram 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 each of 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 of the graphs of $y=x^3-x^2-2x+4$ and $y=4x+4$.

Find the $y$-coordinates of these points of intersection.

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

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

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

Using your graphs, or otherwise, solve the simultaneous equations $y=(x+3)(x-1)$ and $x-y=1$.

Sketch the graph of $y=3x^3-2x^2-x$, giving the coordinates of the points where the graph meets the coordinate axes.

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

Use your diagram to determine the number of solutions of the equation $x^3-2x^2-8x=\frac{1}{x}$.

A machine completes a calculation in a 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 $p$ and $t$.

How many processes are involved in a calculation taking $0.2$ seconds?

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

The diagram shows the graph of $y=\frac{a}{x^2}$, 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 $(m,m^4)$.

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 $y=(x-1{)}^2$ and $y=2-x^2-x$.

Using your graphs, or otherwise, find the solutions of 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 completes a calculation in a time, $t$ seconds, that is proportional to the cube root of the number of processes, $p$, involved. For a calculation involving $8$ processes the machine takes $6.4\times {10}^{-4}$ seconds.

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

Find the time it takes for the machine to complete 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 $(m,m^6)$.

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