4.08 Using quadratic functions (Extended)
Finding the equation of a quadratic
Often, the equation of a quadratic function may need to be found given certain information about the graph or the context. When finding the equation of a line or graphing quadratics, depending on the information given, a particular form might be more convenient. Here is a summary of the forms discussed:
General Form: $y=ax^2+bx+c$y=ax2+bx+c
Factored or $x$x-intercept form: $y=a\left(x-\alpha\right)\left(x-\beta\right)$y=a(x−α)(x−β)
Turning point form: $y=a\left(x-h\right)^2+k$y=a(x−h)2+k
Each of these forms has three unknowns. The method of finding the quadratic equation in each form depends on the information provided:
- To find the equation of a graph in turning point form requires the turning point plus one additional point to find the value of $a$a.
- To find an equation in factored form requires the two $x$x-intercepts and an additional point to find the value of $a$a.
- To find an equation in general form, find the equation in one of the previous forms, depending on information given, then expand to general form. Or, use three points and set up simultaneous equations to solve for $a$a, $b$b and $c$c.
Practice questions
Question 1
A parabola of the form $y=\left(x-h\right)^2+k$y=(x−h)2+k is symmetrical about the line $x=2$x=2, and its vertex lies $6$6 units below the $x$x-axis.
Determine the equation of the parabola.
Graph the parabola.
Loading Graph...
Question 2
Determine the equation of a parabola whose $x$x-intercepts are $-10$−10 and $4$4, and whose $y$y-intercept is $-40$−40.
Express the equation in the following form, for some value of $a$a.
$y=a\left(x+\editable{}\right)\left(x-\editable{}\right)$y=a(x+)(x−)
Note: you do not need to find the value of $a$a at this point.
Hence determine the value of $a$a.
Hence state the equation of the parabola in the form $y=x^2+\ldots$y=x2+…
Applications of quadratics
Previously, the features of quadratic functions and their graphs were explored. Understanding these features allows for some physical situations to be modelled by quadratic equations.
Perhaps the most common real-world occurrence of a quadratic equation is the motion of a falling object. When a ball is thrown at an angle, or hit it with a bat, the path it takes as it falls back to the ground is very close to a parabola (unless it is affected by a strong wind).
A thrown ball follows the path of a parabola as it falls back to the ground.
Quadratic functions of the form $y=ax^2+bx+c$y=ax2+bx+c that model objects falling under gravity may use $x$x to represent the horizontal position of the object. Other times, $x$x or $t$t may represent the time taken to reach a particular height $y$y. Make sure to pay attention to what the variables in the model represent!
The area formulas of various shapes can also be modelled by quadratic functions. For example, the area of a square of length $x$x is given by $A=x^2$A=x2, and the area of a circle of radius $r$r is given by $A=\pi r^2$A=πr2.
There are many other situations that can be modelled by quadratic equations. The braking distance $d$d of a car is related to its speed $x$x by a quadratic equation of the form $d=ax^2+bx$d=ax2+bx.
Braking distance is the distance a vehicle travels between the time the driver applies the brake and the time the vehicle stops.
Modelling physical phenomena
When using an equation to model a physical situation, the context is important when interpreting the results. For example, when modelling a physical situation with a parabola, the context can give meaning to the features of the parabola, such as the intercepts and the turning point.
Think about the example of throwing a ball. This can be modelled using an equation of the form $y=ax^2+bx+c$y=ax2+bx+c, where $y$y is the height of the ball at time $x$x. In this model:
- the $y$y-intercept of the parabola represents the height that the ball was thrown from,
- the turning point represents the maximum height that the ball reaches, and
- the $x$x-intercept represents how long it took to reach the ground.
When modelling the path of a ball using a parabola, the important features can be highlighted, as shown below. Here, restrict the curve to positive times and heights, because only these values exist within this context.
The parts of the parabola drawn with dashes do not model the physical situation of throwing a ball.
When mathematics is used to model real contexts, always consider any practical limitations. For example, the area of a square only makes sense when the side length is positive (that is, for $x>0$x>0), even though the parabola $A=x^2$A=x2 that models it can take negative values.
Practice questions
Question 3
In a game of tennis, the ball is mistimed and hit high up into the air. Initially (ie at $t=0$t=0), the ball is struck $3.5$3.5 metres above the ground and hits the ground $7$7 seconds later. It reaches its greatest height $3$3 seconds after being hit.
Determine the coordinates of the points labelled A, B and C in the diagram.
A $\left(\editable{},\editable{}\right)$(,)
B $\left(\editable{},\editable{}\right)$(,)
C $\left(\editable{},\editable{}\right)$(,)
Using the form of the parabola $y=a\left(t-h\right)^2+k$y=a(t−h)2+k, where $t$t is the number of seconds after the ball is hit and $y$y is the height of the ball above the ground, determine the value of $h$h.
By substituting $t=0$t=0 and $y=3.5$y=3.5 into the equation, form an algebraic relationship between $a$a and $k$k.
By substituting $t=7$t=7 and $y=0$y=0 into the equation, form another algebraic relationship between $a$a and $k$k.
Solve for the exact value of $a$a.
Hence find the exact value of $k$k, the greatest height reached by the ball.
Question 4
A rectangular enclosure is to be built for an animal. Zookeepers have $26$26 metres of fencing, but they want to maximise the area of the enclosure.
Let $x$x be the width of the enclosure.
Form an expression for the length of the enclosure in terms of $x$x.
Form an expression for $A$A, the area of the enclosure.
The area function $A=x\left(13-x\right)$A=x(13−x) has been graphed.
Loading Graph...What width will allow for the greatest possible area?
Width $=$=$\editable{}$ metres
What is the greatest possible area of such an enclosure?
Using technology
Once the important information has been extracted from a question in context and formed into an equation, we could use technology to help graph the equation and determine key features. Don't forget to interpret any result in the context of the problem.
Practice question
Question 5
An object launched from the ground has a height (in metres) after $t$t seconds that is modelled by the equation $y=-4.9t^2+58.8t$y=−4.9t2+58.8t.
Graph this equation using a calculator or other technology then answer the following questions.
What is the maximum height of the object?
After how many seconds is the object at its maximum height?
After how many seconds does the object return to the ground?