IGCSE Math B 4.J Graphs of equations Topic Practice

Graphs and graphical treatment of the equation: EFy Ax Bx Cx D xx= + + +++ in which the constants are numerical and at least three of them are zero Students will be expected to draw and interpret graphs from given equations Use of the intersection of two curves (graphs) to solve equations

[Maximum number: 5]

Part of the curve C with equation $y=-\frac{1}{2} x^{2}+x+\frac{7}{2}$ is drawn on the grid.

(a)

By drawing a suitable straight line on the grid, find estimates, to one decimal place, for the solutions of the equation

-\frac{1}{2} x+1+\frac{1}{2} x^{-1}=0

The equation of curve D is given by

y=x^{3}+x^{2}-3 x

(b)

Complete the table of values for $y=x^3+x^2-3x$ , giving your values to 2 decimal places where necessary.

[ 2 ]

(c)

On the grid opposite, plot the points from your completed table and join them to form a smooth curve.

Curve C and curve D intersect twice in the range $-2\leqslant x\leqslant2$ .

[ 2 ]

(d)

Write down the coordinates, to one decimal place, of these 2 points of intersection.

[ 1 ]

[Maximum number: 3]

A particle P is moving along a straight line through the fixed point O.
The speed, v m/s, of P at time t seconds is given by
$v=\frac{t^2}{27}+2+\frac{3}{t^2}$
for $1\leqslant t\leqslant5$ .

(a)

On the grid opposite, plot the points from your completed table and join them to form a smooth curve.

[ 3 ]

(b)

Using your curve, find an estimate, to one decimal place, for the speed of P when t=1.75.

[Maximum number: 2]

Figure 2

Diagram NOT accurately drawn.

Figure 2 shows a right circular cylinder cut from a solid sphere of radius 4 cm. The height of the cylinder is $2h\text{ cm}$ .
The volume of the cylinder is $V\text{ cm}^3$ .

Part of the curve with equation $V=32\pi h-2\pi h^3$ is drawn on the grid.
Find an estimate, to one decimal place, of each of the two values of h for which the volume of the cylinder is $100\text{ cm}^3$ .

[Maximum number: 7]

Use the grid opposite for parts (b), (c), and (d).

(a)

Complete the table of values for $y=x^2-4x-4$ .

[ 2 ]

(b)

On the grid opposite, plot the points from your completed table and join them to form a smooth curve.

[ 2 ]

(c)

By drawing a suitable straight line on your grid, find estimates to one decimal place for the solutions of the equation $x^2-4x-4=-2x+1$ in the range $-2\leqslant x\leqslant5$ .

[ 3 ]

[Maximum number: 9]

The curve has equation
$y=2x^3-9x^2+3x+7.$

(a)

Complete the table of values for $y=2x^3-9x^2+3x+7$ .

[ 2 ]

(b)

On the grid opposite, plot the points from your completed table and join them to form a smooth curve.

[ 3 ]

(c)

Use your graph to estimate, to one decimal place, the solutions of the equation
$2x^3-9x^2+3x+7=0.$

(d)

By drawing a suitable straight line on the grid, find an estimate for the range of values of x, to one decimal place, in the interval $0<x\leqslant4$ for which
$4x-18+\frac{38}{x^2}<0.$
Show clear algebraic working.

[ 4 ]

[Maximum number: 6]

The graph of
$y=\frac14x^2-3-\frac1x,\quad x\ne0,$
is to be drawn for $-4\leq x\leq4$ .

(a)

Complete the table of values, giving your answers to 2 decimal places.

[ 2 ]

(b)

On the grid, plot the points from your completed table and join them to form a smooth curve.

(c)

Use your graph to find estimates for the solutions of the equation
$\frac14x^2-3-\frac1x=0.$

(d)

By drawing a suitable line on the grid, solve the inequality
$\frac14x^2-2-\frac1x\leq0.$

[ 4 ]

[Maximum number: 7]

The equation of a curve is
$y=5-\frac{x}{2}-x^2.$

(a)

Complete the table of values for
$y=5-\frac{x}{2}-x^2.$

[ 2 ]

(b)

On the grid opposite, plot the points from your completed table and join them to form a smooth curve.

[ 2 ]

(c)

Only use this grid if you need to redraw your curve.

By drawing a suitable straight line on the grid, find estimates, to one decimal place, for the solutions of the equation
$3-x-x^2=0.$

Only use this grid if you need to redraw your curve.

[ 3 ]

[Maximum number: 3]

Diagrams NOT accurately drawn.

Figure 3 shows a rectangle with dimensions 20 cm by 10 cm from which a square with sides of length x cm is removed from each of the corners.
The shape in Figure 3 is then folded along the dotted lines to form a box, without a lid, in the shape of a cuboid, shown in Figure 4.
The volume of the box is $V\,\mathrm{cm}^3$ .

(a)

Complete the following table of values for $V=4x^3-60x^2+200x$ .

(b)

On the grid opposite, plot the points from your completed table and, using your answer to part (b), join them to form a smooth curve.

(c)

Starting with a square of side 15 cm and removing a square with sides of length x cm from each corner, a second box without a lid is formed by folding as in part (a).
The volume of this box is $B\,\mathrm{cm}^3$ , where
$B=4x^3-60x^2+225x.$
Given that B=200, find, by drawing a suitable straight line on the grid, estimates, to one decimal place, of the possible values of x.

[ 3 ]

[Maximum number: 8]

$f(x)=x^3-3x-2$

(a)

Complete the table of values for $y=x^3-3x-2$ .
Give your values of y to one decimal place where necessary.

[ 2 ]

(b)

On the grid opposite, plot the points from your completed table and join them to form a smooth curve.

[ 3 ]

(c)

By drawing a suitable straight line on the grid, find an estimate, to one decimal place, for the solution of the equation
$2x^3-7x-6=0$
in the interval $-2\leqslant x\leqslant2.5$ .

[ 3 ]

[Maximum number: 10]

The curve C has equation
$y=x^2+\frac{24}{x}-25,\quad 0<x\leqslant5.$

(a)

Complete the table of values for y.
Give your values of y to one decimal place where necessary.

[ 2 ]

(b)

On the grid opposite, plot the stationary point and plot the points from your completed table. Join these to form a smooth curve.

[ 4 ]

(c)

By drawing a suitable straight line on the grid, find estimates, to one decimal place, of the solutions to the equation
$x^3+8x^2-49x+24=0$
within the range $0<x\leqslant5$ .

[ 4 ]

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