Further Graphs & Tangents (Cambridge (CIE) O Level Maths): Exam Questions

$\mathrm{f}\left(x\right)=x(x+2)(x–3)$

On the diagram, sketch the graph of $y=\mathrm{f}\left(x\right)\mathrm{for}–3⩽x⩽4$ . Show the values of the intersections with the axes. 

Expand and simplify.

$x(x+2)(x–3)$

The diagram shows the graph of $y=\mathrm{f}\left(x\right)$ for  $-3\le x\le 3$.

i) Solve $\mathrm{f}\left(x\right)=14.$

$x=$................................................ [1]

ii) By drawing a suitable tangent, find an estimate of the gradient of the graph at the point $(-2,4)$.

[3]

By drawing a suitable straight line on the grid, solve$\mathrm{f}\left(x\right)=2x-2$ for $-3\le x\le 3$.

$x=$................................................

The table shows some values for $y=2x^3-4x^2+3$.

Complete the table.

Draw the graph of $y=2x^3-4x^2+3$ for $-1\le x\le 2$.

i) Use your graph to solve the equation $2x^3-4x^2+3=1.5$.

$x=$ ........................ or $x=$ ........................ or $x=$ ........................ [3]

ii) The equation $2x^3-4x^2+3=k$ has only one solution for $-1\le x\le 2$.
Write down a possible integer value of $k$.

[1]

$\mathrm{f}\left(x\right)=\frac{x^2}{4}-\frac{4}{x}$, $x\ne 0$  
Complete the table for $\mathrm{f}\left(x\right)$.

The graph of $y=\mathrm{f}\left(x\right)$ for $-6\le x\le -0.5$ is drawn on the grid.

On the same grid, draw the graph of $y=\mathrm{f}\left(x\right)$ for $0.5\le x\le 6$.

By drawing a suitable tangent, estimate the gradient of the graph of $y=\mathrm{f}\left(x\right)$ at the point (–4, 5).

$\mathrm{g}\left(x\right)=\frac{9}{x}$,  $x\ne 0$  
Complete the table for $\mathrm{g}\left(x\right)$.  

On the same grid, draw the graph of  $y=\mathrm{g}\left(x\right)$ for  $-4\le x\le -1$  and  $1\le x\le 4$.

i) Use your graphs to find the value of $x$ when  $\mathrm{f}\left(x\right)=\mathrm{g}\left(x\right)$.

$x$ = ................................................. [1]

ii) Write down an inequality to show the positive values of $x$ for which  $\mathrm{f}\left(x\right)>\mathrm{g}\left(x\right)$.

 [1]

i) $y=2^x$ Complete the table

  [2]

ii) $y=14-x^2$ Complete the table

  [2]

On the grid, draw the graphs of  $y=2^x$ and  $y=14-x^2$  for  $0\le x\le 4$.

Use your graphs to solve the equations.

i) $2^x=12$

$x=$............................................... [1]

ii) $2^x=14-x^2$

$x=$............................................... [1]

i) On the grid, draw the line from the point $(4,2)$ that has a gradient of $-4$.

[1]

ii) Complete the statement.

This straight line is a .................................. to the graph of $y=14-x^2$ at the point ( ....... , ....... ). [2]

The diagram above shows a sketch of the graph of $y=5\times 3^x+7$.

The graph of the curve crosses the $y$-axis at point $A$.

Find the coordinates of point $A$.

Write down the equation of the asymptote to the graph of $y=5\times 3^x+7$.

$\mathrm{f}\left(x\right)=\frac{20}{x}+x,x\ne 0$ Complete the table.

On the grid, draw the graph of  $y=\mathrm{f}\left(x\right)$ for $–10⩽x⩽–1.6$ and  $1.6⩽x⩽10$.

Using your graph, solve the equation $\mathrm{f}\left(x\right)=11$.

$x$ = ..................   or   $x$ = .................

$k$ is a prime number and $\mathrm{f}\left(x\right)=k$has no solutions.

Find the possible values of $k$.

The gradient of the graph of $y=\mathrm{f}\left(x\right)$ at the point$(2,12)$is $-4$.

Write down the coordinates of the other point on the graph of $y=\mathrm{f}\left(x\right)$ where the gradient is $-4$.

 (....................... , .......................)

i) The equation $\mathrm{f}\left(x\right)=x^2$ can be written as $x^3+p{x}^2+q=0.$ Show that $p=-1$and $q=-20$.

[2]

ii) On the grid, draw the graph of  $y=x^2$ for  $–4⩽x⩽4$.   

[2]

iii) Using your graphs, solve the equation $x^3–x^2–20=0.$

$x$ = .............................................. [1]

iv)

The diagram shows a sketch of the graph of $y=x^3–x^2–20.$
$P$ is the point $(n,0)$.

Write down the value of $n$.

$n=$ .............................................. [1]

The table shows some values for $y=x^3+3x^2+2.$

Complete the table.

On the grid, draw the graph of $y=x^3+3x^2+2$ for $-3.5\le x\le 1.5.$

Use your graph to solve the equation $x^3+3x^2+2=0$ for $-3.5\le x\le 1.5.$  

$x$ = ....................................................

By drawing a suitable straight line, solve the equation $x^3+3x^2+2x+2=0$ for $-3.5\le x\le 1.5.$  

$x$ = ....................................................

For $-3.5\le x\le 1.5,$ the equation $x^3+3x^2+2=k$ has three solutions and $k$ is an integer.

Write down a possible value of $k$.  

$k$ = ....................................................

The table shows some values of  $y=x^3-3x-1$.

  Complete the table of values.

Draw the graph of  $y=x^3-3x-1$  for  $-3\le x\le 3$.

A straight line through (0, –17) is a tangent to the graph of  $y=x^3-3x-1$.

i) On the grid, draw this tangent.    

[1]

ii) Find the co-ordinates of the point where the tangent meets your graph.   

(................ , ................) [1]

iii) Find the equation of the tangent.
Give your answer in the form  $y=mx+c$.  

$y$ = ................................................ [3]

By drawing a suitable straight line on the grid, solve the equation  $x^3-6x-3=0$.  

$x$ = ................... or $x$ = ................... or $x$ = ...................

The diagram shows a circle centre $C(0,1)$. $P(3,5)$is a point on the circumference of the circle.
Find the equation of the tangent at $P$.