GCSE Cubic and reciprocal graphs Practice Questions

Write down the equation

The equation is the rule that turns each value of x into a value of y.

Substitute $x = 3$

Replace every x with 3. Keep the brackets so that the power is worked out before the adding and subtracting.

Work out the value of y

3 cubed means 3 × 3 × 3, which is 27.

Recall what a reciprocal graph looks like

A reciprocal equation has x on the bottom of a fraction.

Check each equation

$y = 8x$ is a straight line, $y = x^3$ is a cubic, $y = x^2$ is a parabola and $y = 8 - x$ is a straight line. Only $y = 8/x$ has x in the denominator.

Choose the reciprocal equation

$y = 8/x$ is the reciprocal graph: two branches, never touching either axis.

Write down the equation

t is found by dividing 60 by x.

Substitute $x = 24$

Replace x with 24 in the denominator.

Write the fraction as a division

A fraction bar means divide, so the top number is divided by the bottom number.

Work out the division

$60 \div 24 = 2.5$.

Check the sign

Both numbers have the same sign, so the answer is positive.

State the value

So when $x = 24$, $t = 2.5$.

Write down the equation

y is 1 divided by x.

Substitute $y = -0.2$

You are told the y-value and need the matching x-value.

Multiply both sides by x

Multiplying by x clears the fraction; $x = 0$ is not allowed anyway.

Divide both sides by -0.2

Divide by the coefficient of x.

Write -0.2 as a fraction

Turning the decimal into a fraction makes the division easier.

Divide by the fraction

Dividing by one fifth is the same as multiplying by 5, and the sign is negative.

Check the answer

Substituting $x = -5$ gives $y = -0.2$, as required.

Check the sign

On $y = 1/x$ a negative y-value can only come from a negative x-value.

Identify the branch

This point is on the branch in the third quadrant, where x and y are both negative.

State the answer

The required value of x is -5.

Write down the equation

The solutions are the x-values where the curve crosses the x-axis.

Try $x = 1$

This is not 0, so $x = 1$ is not a solution.

Try $x = 2$

The value is 0, so $x = 2$ is a solution and (x - 2) is a factor.

Divide by (x - 2)

Dividing the cubic by the known factor leaves a quadratic.

Check the division by expanding

Expanding gives back the original cubic.

Factorise the quadratic

5 × (-3) = -15 and 5 + (-3) = 2.

Write the cubic fully factorised

The cubic is a product of three linear factors.

Set each factor equal to zero

A product is zero only when one of the factors is zero.

Solve each equation

These are the three solutions.

Put them in order

Ordering the solutions makes it easy to pick the smallest.

Choose the smallest solution

The smallest of the three solutions is -5.

Check $x = -5$

Substituting $x = -5$ makes the equation true.

Check $x = 3$

This root works as well, so the factorisation is right.

Interpret on the graph

The cubic crosses the x-axis at three points, the leftmost being (-5, 0).

State the answer

The smallest solution is -5.