A quadratic model describes a quantity that rises to a greatest value and then falls, or falls to a least value and then rises: a height, an area, a revenue or the curve of an arch. Write the model in completed-square form y=a(x-h)2+k; its features answer most questions. **Feature** | **Meaning in the context** Turning point (h,,k) | Greatest value k when a0, reached when the input is h. This value k is also called the stationary value: the value of the function at its turning point. Value when the input is 0 | Starting value, for example the height of a nozzle or a fixed cost Roots of y=0 | When the height is zero (lands), or where an arch meets the ground Line of symmetry x=h | The highest (or lowest) point is halfway between any two points at the same height, such as the two feet of an arch or the two roots Why: a fountain's water jet leaves a nozzle 2 m above the surface of a pool. Its height is y m above the water at a horizontal distance of x m from the nozzle, where y=2+12x-4x2, a downward parabola. Completing the square, y=-4(x-1.5)2+11: the squared term is never negative, so y 11, with equality only when x=1.5. The model only applies from the nozzle, x=0, to where the jet meets the water. Method - Read the model, or form it: name each variable with its unit, write one equation from the given fact (a total length, a perimeter, a price rule) and substitute. - State the values the context allows, for example lengths greater than 0 or times from release until landing. - Complete the square to find the greatest or least value and the input at which it occurs. - To find when the model takes a given value, solve the equation and reject any solution outside the allowed values, saying why. - Answer in words with units; give non-exact answers to 3 significant figures. Example A shop sells caps at $x each and expects to sell 60-2x caps a day. Its daily revenue is R=x(60-2x)=-2x2+60x=-2[(x-15)2-225]=-2(x-15)2+450. The greatest revenue is $450, at a price of $15. For the jet above, it meets the water when y=0: (x-1.5)2= 114, so x=1.5+- sqrt(11)2. Reject x=-0.158, which is behind the nozzle at x=0. The jet meets the water 3.16 m from the nozzle (3 s.f.). Its highest point, at x=1.5, is not halfway to 3.16 m, because the jet starts 2 m above the water. An "explain" or "justify" answer gives the deciding value and compares it with the requirement in one sentence: "The greatest height is 11 m, which is less than 12 m, so the jet cannot reach a sign 12 m above the water." **Exam check** Every root of the equation must be checked against the context: a negative time or length is rejected with a reason, not left out silently. State units in the final answer.

Quadratic Models

Secondary 3

A quadratic model describes a quantity that rises to a greatest value and then falls, or falls to a least value and then rises: a height, an area, a revenue or the curve of an arch. Write the model in completed-square form $y=a(x-h)^2+k$; its features answer most questions.

Feature	Meaning in the context
Turning point $(h,\,k)$	Greatest value $k$ when $a<0$, least value $k$ when $a>0$, reached when the input is $h$. This value $k$ is also called the stationary value: the value of the function at its turning point.
Value when the input is $0$	Starting value, for example the height of a nozzle or a fixed cost
Roots of $y=0$	When the height is zero (lands), or where an arch meets the ground
Line of symmetry $x=h$	The highest (or lowest) point is halfway between any two points at the same height, such as the two feet of an arch or the two roots

Why: a fountain's water jet leaves a nozzle $2$ m above the surface of a pool. Its height is $y$ m above the water at a horizontal distance of $x$ m from the nozzle, where $y=2+12x-4x^2$, a downward parabola. Completing the square, $y=-4(x-1.5)^2+11$: the squared term is never negative, so $y\le11$, with equality only when $x=1.5$. The model only applies from the nozzle, $x=0$, to where the jet meets the water.

Method

Read the model, or form it: name each variable with its unit, write one equation from the given fact (a total length, a perimeter, a price rule) and substitute.
State the values the context allows, for example lengths greater than $0$ or times from release until landing.
Complete the square to find the greatest or least value and the input at which it occurs.
To find when the model takes a given value, solve the equation and reject any solution outside the allowed values, saying why.
Answer in words with units; give non-exact answers to 3 significant figures.

Example

A shop sells caps at $\$x$ each and expects to sell $60-2x$ caps a day. Its daily revenue is $R=x(60-2x)=-2x^2+60x=-2\left[(x-15)^2-225\right]=-2(x-15)^2+450$. The greatest revenue is $\$450$, at a price of $\$15$.

For the jet above, it meets the water when $y=0$: $(x-1.5)^2=\dfrac{11}{4}$, so $x=1.5\pm\dfrac{\sqrt{11}}{2}$. Reject $x=-0.158$, which is behind the nozzle at $x=0$. The jet meets the water $3.16$ m from the nozzle (3 s.f.). Its highest point, at $x=1.5$, is not halfway to $3.16$ m, because the jet starts $2$ m above the water.

An "explain" or "justify" answer gives the deciding value and compares it with the requirement in one sentence: "The greatest height is $11$ m, which is less than $12$ m, so the jet cannot reach a sign $12$ m above the water."

Exam check Every root of the equation must be checked against the context: a negative time or length is rejected with a reason, not left out silently. State units in the final answer.

Create a free account to practise similar questions.

Create free account Sign in Get the A Math course · from S$39.98/mo A Math tuition

Need help? Join our A-Maths tuition classes.

Learn more

Feature	Meaning in the context
Turning point \((h,\,k)\)	Greatest value \(k\) when \(a<0\), least value \(k\) when \(a>0\), reached when the input is \(h\). This value \(k\) is also called the stationary value: the value of the function at its turning point.
Value when the input is \(0\)	Starting value, for example the height of a nozzle or a fixed cost
Roots of \(y=0\)	When the height is zero (lands), or where an arch meets the ground
Line of symmetry \(x=h\)	The highest (or lowest) point is halfway between any two points at the same height, such as the two feet of an arch or the two roots