Hard GCSE Quadratic graphs Questions

At the meeting points the y-values are equal

Set the curve equal to the line.

Add $9$ to both sides

Collect the numbers on the right.

Take the square root

Remember both the positive and the negative root.

Find the first point

Both points lie on the line $y = 7$.

Find the second point

The second intersection.

Check the first point on the curve

The curve does have height $7$ at $x = -4$.

Check the second point

And also at $x = 4$.

Note that the points are level

Both have the same y-coordinate, so the distance between them is horizontal.

Find the horizontal distance

Subtract the x-coordinates.

Check with the line of symmetry

The curve is symmetrical about the y-axis, so the points are $4$ units either side of it.

Confirm the distance

Twice the distance from the axis of symmetry.

Find the turning point

The minimum of the curve is at ($0$, $-9$).

Check the line is above the minimum

The line is above the turning point, so it must cut the curve twice.

Interpret the answer

The two points are $8$ units apart.

State the answer

The distance between the two points is $8$ units.

Use the roots to write the factors

Roots at $-2$ and $3$ give these factors, with a stretch factor a.

Use the y-intercept

Substitute this point into the equation.

Substitute $x = 0$

Replace x with 0 and y with 12.

Work out the brackets

$2$ times $-3$ is $-6$.

Form an equation in a

Substitute the value of the product.

Solve for a

Divide both sides by $-6$.

Interpret the sign of a

The parabola is n-shaped, which fits: it is above the axis between its roots, where the y-intercept $12$ lies.

Expand the brackets

$x^2 - 3x + 2x - 6 = x^2 - x - 6$.

Multiply by a

Multiply each term by $-2$, changing every sign.

Write the equation

This is the required form.

Check the y-intercept

The curve passes through ($0$, $12$).

Check the first root

$x = -2$ gives $y = 0$.

Check the second root

$x = 3$ also gives $y = 0$.

Find the line of symmetry

This is halfway between the roots $-2$ and $3$, as expected.

State the equation

This parabola has all three required features.

Draw the line $y = 2$

The solutions are where the curve reaches a height of $2$.

Set the expression equal to $2$

This is the equation to solve.

Make one side zero

Subtract $2$ from both sides.

Check for whole-number factors

The quadratic does not factorise, so the solutions are not whole numbers.

Find the line of symmetry

The two solutions are equally spaced either side of $x = 2.5$.

Work out the rearranged expression at $x = 0$

The rearranged curve is above the axis here.

Work out the value at $x = 1$

Now it is below the axis, so a solution lies between $x = 0$ and $x = 1$.

Try $x = 0.4$

Just above zero.

Try $x = 0.5$

Just below zero, so the solution lies between $0.4$ and $0.5$.

Refine

Very close to zero, so the smaller solution is about $0.44$.

Use symmetry for the other solution

The solutions are equally spaced about $x = 2.5$.

Check the larger solution

Also very close to zero, so $x = 4.56$ is the other solution.

Check on the original curve

The height is $1.99$, which rounds to $2$, as required.

Round the smaller solution

$0.44$ rounds to $0.4$ to $1$ decimal place.

Round the larger solution

$4.56$ rounds to $4.6$ to $1$ decimal place.

Decide the shape

The coefficient of $x^2$ is negative, so the graph is n-shaped.

A profit means P is above the horizontal axis

The stall makes a profit when the curve is above the x-axis.

Find where the curve meets the axis

Set $P = 0$.

Multiply through by $-1$

This makes the quadratic easier to factorise.

Look for two numbers

They multiply to $16$ and add to $-10$: these are $-2$ and $-8$.

Factorise

Check by expanding.

Solve the first bracket

The curve crosses the axis at $x = 2$.

Solve the second bracket

It crosses again at $x = 8$.

Think about the n-shape

An n-shaped curve is above the axis between its two roots and below it outside them.

Test a price between the roots

At a price of £$5$ the profit is £$9$, which is positive.

Test a price below the smaller root

At £$1$ the stall makes a loss of £$7$.

Test a price above the larger root

At £$9$ the stall also makes a loss of £$7$.

Find the best price

The maximum profit is at the turning point, $x = 5$.

Work out the maximum profit

The greatest profit is £$9$, at a price of £$5$.

Write the range of prices

The stall makes a profit for any price strictly between £$2$ and £$8$.

Decide the shape

The coefficient of $t^2$ is negative, so the graph is n-shaped with a maximum.

The greatest height is at the turning point

The maximum lies on the line of symmetry.

Substitute $a = -1$ and $b = 6$

The ball is highest after $3$ seconds.

Substitute $t = 3$ into the equation

Square first, then multiply.

Work out the squared term

Take care with the negative square.

Work out the height

The greatest height is $9$ metres.

Check the height $1$ second earlier

At $t = 2$ the ball is at $8$ metres.

Check the height $1$ second later

At $t = 4$ it is also at $8$ metres, equal by symmetry about $t = 3$.

Confirm the maximum

Both neighbouring heights are lower, so 9 m is the greatest.

Find when the ball is at ground level

Set $h = 0$.

Factorise

Take out a common factor of -t.

Solve

The ball leaves the ground at $t = 0$ and lands at $t = 6$.

Check the symmetry of the flight

The highest point is halfway through the flight, at $t = 3$, as found.

Interpret the graph

The turning point of the graph is ($3$, $9$).

State the answer

The greatest height reached by the ball is $9$ metres.