02.08 Quadratic Modelling
In this lesson you will learn how to:
- Use quadratic models to solve real-life problems and make predictions
- Give the physical interpretation of the values of constants in a quadratic model
- Investigate the validity of a model in the long-term
- Form a quadratic model when given information about the related variables
Prerequisites from previous chapters:
- Expanding Brackets
- Factorising
VIDEO LESSON
INTERACTIVE SELF-STUDY
A mathematical model describes real-life situations using mathematical concepts to represent patterns, relationships, and predict outcomes.
These models can range from simple to complex, with results varying from approximate to exact, and are often valid only under specific conditions. In Statistics and Mechanics, you’ll learn about the simplifications and assumptions behind these models.
Quadratic functions, for example, are used to explore practical scenarios like projectile motion.
As a recap:
i. How do we find the values of \(x\) for which a quadratic function \(f(x)=0\)? Provide two methods.
iii. What can you use to determine whether a quadratic function \(f(x)\) actually has roots and, if so, how many?
INTERPRETING CONSTANTS AND INVESTIGATING VALIDITY
You may be asked for the physical interpretation of the constants in a quadratic model.
iv. Let \(f(x)=ax^2+bx+c\). What does \(c\) represent graphically? What does it tell you about \(f(x)\) and \(x\)?
- For a quadratic model \(f(x)=ax^2+bx+c\), \(c\) represents the value of \(f\) when \(x=0\).
- If the independent variable \(x\) represents time, you can say \(c\) represents the initial value of \(f\)
You must reword this so that is in the context of the model
You may be asked why a quadratic model may not be valid for large values of \(x\)
v. What happens to the value of \(f(x)=ax^2+bx+c\) as \(x→+∞\) if \(a>0\)? Why might this not be realistic?
- For any piece of information given, ask yourself: “What does this tell me about the value of the variables?”
You should be able to interpret it as an equation or inequality
e.g. “when \(x=\)…, \(y=\)…” , or “find \(x\), when \(y=\)…” - Solutions to any equations should make physical sense. You can use this to determine which solution to an equation is correct.
For example, if you obtain two values for time, but one of them is a negative value then you know to discard that solution in favour of the positive solution. - You may be asked about other limitations or reliability of a model. These questions often require you to give a reasonable, common sense answer that describes other real-life factors that could affect the scenario being modelled.
Example 1 – (GRADE C)
A ball is thrown upwards from the edge of a cliff. The height, in metres, of the ball above the ground after \(t\) seconds is modelled by the function:
\(h(t)=25+20t−5t^2\), \(t>0\)
a. Interpret the meaning of the constant term \(25\) in the model.
c. After how many seconds does the ball hit the ground?
d. Explain why the model is not valid for large values of \(t\)
e. Using your answer to part c, state the range of values of \(t\) for which the model is valid.
f. Suggest one reason why this quadratic model may not be able to model the ball’s height exactly/reliably.
g. Write \(h(t)\) in the form \(A−B(t−C)^2\), where \(A, B\), and \(C\) are constants to be found.
h. Using your answer to part g or otherwise, find the maximum height of the ball above the ground, and the time at which this maximum height is reached.
Example 2 – (GRADE C)
An orchard owner is planning to plant a new apple field.
Let the number of trees in the field be \(T\) and the number of apples per tree be \(N\)
They know from past experience that if they plant \(40\) trees in the field, they will get a yield of \(350\) apples per tree.
They also know that for every \(1\) extra tree they plant, the number of apples per tree decreases by \(5\).
Therefore the number of apples per tree is given by \(N=k-5T\), where \(k\) is a constant.
a. Find the value of \(k\)
b. Hence find an expression for the total yield of apples, \(Y\), in terms of \(T\)
c. Using your model from part b, find the number of trees the orchard owner should plant to maximise the total yield.
d. Find the possible number of trees in the field for which no apples are produced.
FINDING THE EQUATION OF A QUADRATIC MODEL
Often information given in the question can be interpreted as coordinates of points that lie on a quadratic curve. Given this information, you may have to determine the unknown coefficients of the quadratic equation of the model \(f(x)=ax^2+bx+c\).
To help you, you should plot these points on an axes and sketch a graph. You could either be given:
- Two roots and some other point
Recall that \(f(x)=ax^2+bx+c\) can (sometimes) be expressed in factorised form: \(f(x)=a(x−α)(x−β)\) where \(α\) and \(β\) are roots.
So substitute the roots \(α\) and \(β\) into the factorised form, then substitute the last point into the factorised form to solve for \(a\).
If required, you can expand the factorised form to the general form \(ax^2+bx+c\) - The turning point and some other point
Recall that \(f(x)=ax^2+bx+c\) can be expressed in completed square form: \(f(x)=a(x−p)^2+q\) where \((p,q)\) is the turning point.
So substitute the turning point \((p,q)\) into the completed square form, then substitute the last point into the completed square form to solve for \(a\).
If required, you can expand the completed square form to the general form \(ax^2+bx+c\) - Any three random points
Substitute each point into the equation \(y=ax^2+bx+c\) to obtain 3 different equations that \(a,b\) and \(c\) satisfy, then solve for \(a,b\) and \(c\) simultaneously
Example 3 – (GRADE B)
For a certain chemical reaction, \(t\) seconds after the start of the reaction, the temperature of the reaction mixture, \(y\) degrees Celsius (°C), is given by the equation:
\(y=at^2+bt+c\), \(t≥0\) where \(a,b\) and \(c\) are constants.
- At \(t=0\) the temperature is 3 °C
- At \(t=2\) the temperature is 0 °C
- At \(t=6\) the temperature is 0 °C
a. Determine the value of \(a,b\) and \(c\)
b. Find the minimum temperature and the time at which it occurs
c. Comment on the reliability of this model at predicting the temperature long after the reaction has started
Example 4 – (GRADE B)
The ends of a long cable are attached to the top of two vertical posts. The height of the cable above the ground, \(h\) metres, when at a horizontal distance \(x\) metres from the first post is modelled by the equation:
\(h=ax^2+bx+c\), where \(a,b\) and \(c\) are constants
- The height of the first post is \(2\) metres
- The cable is closest to the ground when it is at a horizontal distance of \(5\) metres away from the first post and at a height of \(0.5\) metres
a. Determine the value of \(a,b\) and \(c\)
b. It is given that the second post is the same height as the first post. Find the horizontal distance between the two posts
The information can be interpreted as coordinates \((0, 2)\), and \((5, \frac{1}{2})\).
c. It is instead given that the second post is 1.44 metres taller than the first post. Find the horizontal distance between the two posts.
Example 5 – (GRADE B)
In case of an emergency, the typical stopping distance of a car, \(d\) metres, when travelling at a speed \(v\) miles per hour is given by:
\(d=av^2+bv+c\), where \(a,b\) and \(c\) are constants.
A typical car takes:
- 12 metres to stop if travelling at 20 miles per hour
- 23 metres to stop if travelling at 30 miles per hour
- 36 metres to stop if travelling at 40 miles per hour.
a. Determine the value of \(a, b\) and \(c\)
b. Explain the physical significance of the value of \(c\)c. By drawing a sketch of \(d=av^2+bv+c\) or using your answer to part b, explain why the model is unreliable for small values of \(v\)
We can also see this when sketching a graph of \(d\) (vertical axis) against \(v\) (horizontal axis).d. Find the speed of car that has a total stopping distance of 183 metres.
END OF LESSON!
You can now answer:
EDEXCEL Pure Year 1 Ex 2G – All questions