Further Graphs & Tangents (Cambridge International IGCSE Mathematics): Exam Questions

The diagram shows the graph of a function.

By drawing a suitable tangent, find an estimate for the gradient of the function at the point P.

i) Draw the tangent to the curve at $x=1$.

ii) Use your tangent to estimate the gradient of the curve at $x=1$.

Write down the equation of your tangent in the form $y=mx+c$.

$y=$ .................................................

By drawing a suitable tangent, estimate the gradient of the curve at the point P.

The graph of $y=10-8x^{2}$ for $-1.5\leq x\leq1.5$ is drawn on the grid.

Write down the equation of the line of symmetry of the graph.

On the grid, draw the tangent to the curve at the point where $x=0.5$.

Find the gradient of this tangent.

The diagram shows part of the graph of $y=x^{2}-2x+3$

By drawing a suitable straight line, use your graph to find estimates for the solutions of $x^{2}-3x-1=0$

P is the point on the graph of $y=x^{2}-2x+3$ where $x=2$

Calculate an estimate for the gradient of the graph at the point P.

Complete the table of values for $y=x^{2}-x-6$

On the grid, draw the graph of $y=x^{2}-x-6$ for values of $x$ from $-3$ to $3$

Use your graph to find estimates of the solutions to the equation $x^{2}-x-6=-2$

Complete the table of values for $y=x^{2}-2x$

On the grid, draw the graph of y = x² – 2x for values of x from –2 to 4

Solve $x^{2}-2x-2=1$

$y=x^{2}+\frac{1}{x},x\neq0$

Complete the table.

Extract:

| x | 0.2 | 0.3 | 0.5 | 1 | 1.5 | 2 | 2.5 | | y | 5.0 | 3.4 | 2.3 | | 2.9 | | 6.7 |

On the grid, draw the graph of $y=x^{2}+\frac{1}{x}$ for $0.2\leq x\leq2.5.$ The graph of $y=x^{2}+\frac{1}{x}$ for $-2.5\leq x\leq-0.2$ has been drawn for you.

Extract:

On the grid, draw the graph of y = x^2 + 1/x for 0.2 ≤ x ≤ 2.5. The graph of y = x^2 + 1/x for −2.5 ≤ x ≤ −0.2 has been drawn for you.

By drawing suitable straight lines on the grid, solve the following equations.

$\,$i) $x^{2}+\frac{1}{x}=-2$

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

$\,$ii)$x^{2}+\frac{1}{x}+x-1=0$

$x=$ ................................................ [2]

$k$ is an integer and the equation $x^{2}+\frac{1}{x}=k$ has three solutions. Write down a possible value of $k$.

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

The table shows some values of $y=\frac{x^{2}}{2}+\frac{1}{x^{2}}-\frac{2}{x}$, $x\neq0$.

Complete the table.

Extract:

The table shows some values of y = x^2/2 + 1/x^2 - 2/x, x ≠ 0.

| x | -3 | -2 | -1 | -0.5 | -0.3 | | 0.2 | 0.3 | 0.5 | 1 | 2 | 3 | | y | 5.3 | 3.3 | | 8.1 | 17.8 | | | 4.5 | 0.1 | -0.5 | 1.3 | | Complete the table.

On the grid, draw the graph of $y=\frac{x^{2}}{2}+\frac{1}{x^{2}}-\frac{2}{x}$ for $-3\leq x\leq-0.3$ and $0.2\leq x\leq3$.

Extract:

On the grid, draw the graph of y = x^2/2 + 1/x^2 - 2/x for -3 ≤ x ≤ -0.3 and 0.2 ≤ x ≤ 3.

Use your graph to solve $\frac{x^{2}}{2}+\frac{1}{x^{2}}-\frac{2}{x}\leq0$.

..................... $\leq x\leq$ ....................

Find the smallest positive integer value of $k$ for which $\frac{x^{2}}{2}+\frac{1}{x^{2}}-\frac{2}{x}=k$ has two solutions for $-3\leq x\leq-0.3$ and $0.2\leq x\leq3$.

Extract:

The table shows some values of y = x^2/2 + 1/x^2 - 2/x, x ≠ 0. | x | -3 | -2 | -1 | -0.5 | -0.3 | | 0.2 | 0.3 | 0.5 | 1 | 2 | 3 | | y | 5.3 | 3.3 | | 8.1 | 17.8 | | | 4.5 | 0.1 | -0.5 | 1.3 | |

By drawing a suitable straight line, solve $\frac{x^{2}}{2}+\frac{1}{x^{2}}-\frac{2}{x}=3x+1$ for $-3\leq x\leq-0.3$ and $0.2\leq x\leq3$.

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

The table shows some values for $y=x^{3}+x^{2}-5x$.

Complete the table.

Extract:

The table shows some values for y = x^3 + x^2 - 5x.

| x | -3 | -2 | -1.5 | -1 | 0 | 1 | 1.5 | 2 | 2.5 | 3 | | y | -3 | 6 | 6.4 | | 0 | | -1.9 | 2 | 9.4 | |

On the grid, draw the graph of $y=x^{3}+x^{2}-5x$ for $-3\leq x\leq3$.

Extract:

On the grid, draw the graph of y = x cubed plus x squared minus 5 x for -3 ≤ x ≤ 3.

Use your graph to solve the equation $x^{3}+x^{2}-5x=0$.

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

Extract:

On the grid, draw the graph of y = x cubed plus x squared minus 5 x for -3 ≤ x ≤ 3.

By drawing a suitable tangent, find an estimate of the gradient of the curve at $x=2$.

Extract:

The table shows some values for y = x^3 + x^2 - 5x.

| x | -3 | -2 | -1.5 | -1 | 0 | 1 | 1.5 | 2 | 2.5 | 3 | | y | -3 | 6 | 6.4 | | 0 | | -1.9 | 2 | 9.4 | |

Write down the largest value of the integer, $k$, so that the equation $x^{3}+x^{2}-5x=k$ has three solutions for $-3\leq x\leq3$.

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

The table shows some values for $y=\frac{3}{10}x^{3}-2x$ for $-3\leq x\leq3$.

Complete the table.

On the grid, draw the graph of $y=\frac{3}{10}x^{3}-2x$ for $-3\leq x\leq3$.

Extract:

On the grid, draw the graph of y = 3/10 x^3 - 2x for -3 <= x <= 3. On the grid, draw a suitable straight line to solve the equation 3/10 x^3 - 2x = 1/2 (1 - x) for -3 <= x <= 3.

On the grid, draw a suitable straight line to solve the equation $\frac{3}{10} x^{3} - 2 x = 1 \frac{1}{2} ( 1 - x )$ for $-3\leq x\leq3.$

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

Extract:

The table shows some values for y = 3/10 x^3 - 2x for -3 ≤ x ≤ 3.

| x | -3 | -2 | -1.5 | -1 | 0 | 1 | 1.5 | 2 | 3 | | y | | | 2.0 | 1.7 | 0 | | -2.0 | -1.6 | | Complete the table.

For $-3\leq x\leq3$, the equation $\frac{3}{10}x^{3}-2x=1$ has $n$ solutions. Write down the value of $n$.

$n=$..................................................

The table shows some values of $y=x^{3}-3x^{2}+x$.

Complete the table.

Extract:

The table shows some values of y = x^3 - 3x^2 + x.

| x | -0.75 | -0.5 | -0.25 | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 2.75 | | y | -2.9 | -1.4 | -0.5 | | -0.1 | -1 | -1.9 | | -0.6 | |

On the grid, draw the graph of $y=x^{3}-3x^{2}+x$ for $-0.75\leq x\leq2.75$.

Extract:

On the grid, draw the graph of y = x^3 - 3x^2 + x for -0.75 ≤ x ≤ 2.75.

Use your graph to complete the inequalities in $x$ for which $y>-1$.

.................... $<x<$ .................... and $x>$ ....................

Extract:

On the grid, draw the graph of y = x cubed minus 3 x squared plus x for -0.75 ≤ x ≤ 2.75.

The equation $x^{3}-3x^{2}+2x-1=0$ can be solved by drawing a straight line on the grid.

i) Write down the equation of this line.

ii)On the grid, draw this line and use it to solve the equation $x^{3}-3x^{2}+2x-1=0$.

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

By drawing a suitable tangent, find an estimate for the gradient of the graph of $y=x^{3}-3x^{2}+x$ at $x=-0.25$.

Complete the table of values for $y=\frac{x^{3}}{3}-\frac{1}{2x^{2}},x\neq0$.

Extract:

Complete the table of values for y = x^3/3 - 1/(2x^2), x ≠ 0.

| x | -3 | -2 | -1 | -0.5 | -0.3 | | 0.3 | 0.5 | 1 | 2 | 3 | | y | -9.1 | -2.8 | -0.8 | | -5.6 | | -5.5 | -2.0 | | | 8.9 |

On the grid, draw the graph of $y=\frac{x^{3}}{3}-\frac{1}{2x^{2}}$ for $-3\leq x\leq-0.3$ and $0.3\leq x\leq3$.

Extract:

y equals x cubed over 3 minus 1 over 2 x squared, x not equal to 0. On the grid, draw the graph of y equals x cubed over 3 minus 1 over 2 x squared for negative 3 less or equal than x less or equal than negative 0.3 and 0.3 less or equal than x less or equal than 3.

i) By drawing a suitable tangent, find an estimate of the gradient of the curve at $x=-2$.

ii) Write down the equation of the tangent to the curve at $x=-2$. Give your answer in the form $y=mx+c$.

$y=$ ................................................ [2]

Extract:

Complete the table of values for y = x^3/3 - (1/2)x^2, x ≠ 0. On the grid, draw the graph of y = x^3/3 - (1/2)x^2 for -3 ≤ x ≤ -0.3 and 0.3 ≤ x ≤ 3.

Use your graph to solve the equations.

i)$\frac{x^{3}}{3}-\frac{1}{2x^{2}}=0$

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

ii)$\frac{x^{3}}{3}-\frac{1}{2x^{2}}+4=0$

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

The table shows some values of $y=\frac{1}{2x}-\frac{x}{4}$ for $0.15\leq x\leq3.5.$

Complete the table.

Extract:

The table shows some values of y = 1/(2x) - x/4 for 0.15 <= x <= 3.5.

| x | 0.15 | 0.2 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | | y | 3.30 | | 0.88 | | -0.04 | | -0.43 | -0.58 | -0.73 |

On the grid, draw the graph of $y=\frac{1}{2x}-\frac{x}{4}$ for $0.15\leq x\leq3.5.$ The last two points have been plotted for you.

Extract:

The table shows some values of y = (1/(2x)) - (x/4) for 0.15 ≤ x ≤ 3.5.

Use your graph to solve the equation $\frac{1}{2x}-\frac{x}{4}=\frac{1}{2}$ for $0.15\leq x\leq3.5.$

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

Extract:

The table shows some values of y = 1/(2x) - x/4 for 0.15 ≤ x ≤ 3.5.

| x | 0.15 | 0.2 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | | y | 3.30 | | 0.88 | | -0.04 | | -0.43 | -0.58 | -0.73 |

i) On the grid, draw the line $y=2-x.$

ii) Write down the x co-ordinates of the points where the line $y=2-x$ crosses the graph of $y=\frac{1}{2x}-\frac{x}{4}$ for $0.15\leq x\leq3.5$.

$x$ = .................... and $x$ = .................... [2]

Extract:

The table shows some values of y = 1/(2x) - x/4 for 0.15 ≤ x ≤ 3.5.

Show that the graph of $y=\frac{1}{2x}-\frac{x}{4}$ can be used to find the value of $\sqrt{2}$ for $0.15\leq x\leq3.5.$

The diagram shows the graph of $y = f ( x )$ where $f ( x ) = x^{2} - \frac{2}{x} - 2 , x \text{eq} 0$.

On the grid, draw a suitable straight line to solve the equation $x^{2}-\frac{2}{x}-7=-3x$ for $-3\leq x\leq3$.

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

Extract:

The diagram shows the graph of y = f(x) where f(x) = (x^2 - 2)/(x - 2), x ≠ 0.

By drawing a suitable tangent, find an estimate of the gradient of the curve at $x=-2$.

Extract:

The diagram shows the graph of y = f(x) where f(x) = (x^2 - 2)/(x - 2), x ≠ 0.

i) Complete the table for $y=\text{g}(x)$ where $\text{g}(x)=2^{-x}$ for $-3\leq x\leq3$.

ii) On the grid, draw the graph of $y=\text{g}(x)$.

iii) Use your graph to find the positive solution to the equation $\text{f}(x)=\text{g}(x)$.

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

The graph of $y=10-8x^{2}$ for $-1.5\leq x\leq1.5$ is drawn on the grid.

And the table shows some values for $y=x^{3}+3x+4$.

i) Complete the table.

ii) On the grid, draw the graph of $y=x^{3}+3x+4$ for $-1.5\leq x\leq x1.5$.

Extract:

The graph of y = 10 - 8x^2 for -1.5 ≤ x ≤ 1.5 is drawn on the grid. And the table shows some values for y = x^3 + 3x + 4.

Show that the values of $x$ where the two curves intersect are the solutions to the equation $x^{3}+8x^{2}+3x-6=0$.

By drawing a suitable straight line, solve the equation $x^{3}+5x+2=0$ for $-1.5\leq x\leq1.5$.

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

The table shows some values for $y=2x+\frac{1}{x}-3$ for $0.125\leq x\leq3.$

Complete the table.

Extract:

The table shows some values for y = 2x + 1/(x - 3) for 0.125 ≤ x ≤ 3.

| x | 0.125 | 0.25 | 0.375 | 0.5 | 0.75 | 1 | 1.5 | 2 | 2.5 | 3 | | y | 5.25 | 1.5 | 0.42 | | | 0 | 0.67 | 1.5 | | 3.33 | Complete the table.

On the grid, draw the graph of $y=2x+\frac{1}{x}-3$ for $0.125\leq x\leq3.$

Extract:

On the grid, draw the graph of y = (2x + 1)/(x - 3) for 0.125 ≤ x ≤ 3.

Use your graph to solve $2x+\frac{1}{x}-3\geq2.$

................................................. .................................................

Extract:

On the grid, draw the graph of y = (2x + 1)/(x - 3) for 0.125 ≤ x ≤ 3.

The equation $\frac{1}{x}=7-3x$ can be solved using your graph in part (b) and a straight line.

i) Write down the equation of this straight line.

ii) Draw this straight line and solve the equation $\frac{1}{x}=7-3x.$

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