Section 4 - Quadratic functions
The topic of quadratic functions has been broken down into the following sections:
- 4A - Expanding quadratics
- 4B - Factorising quadratics
- 4C - Solving quadratic equations
- 4D - The general quadratic formula
- 4E - The discriminant
- 4F - Completing the square
- 4G - Graphing quadratic functions
- 4H - Solving quadratic simultaneous equations
- 4I - Solving quadratic inequations
- 4J - Rules for quadratic graphs
- 4K - Modelling with quadratic functions
- 4L - Exploring rates of change with parabolas
Setting the scene: The pig pen
The area of a rectangle, with a given constraint, can be maximised using knowledge of quadratic functions.
-
The pig pen task
-
Interactive model
-
Solutions
<
>
The adjoining, rectangular pig pens are to be built as shown by the diagram. 40 metres of fencing material is available. The area of the pens is important for the pig's health and well-being.
|
Click here if the GeoGebra applet below does not load properly.
(a)
|
The total length of the fencing sums to 40 metres \[4x+6y=40\] \[\therefore y=\frac{40-4x}{6}\]
|
(b)
|
The area of a single pig pen is given by \(A=x\times y\) \[A(x)=x\times \frac{40-4x}{6}\] \[\therefore A(x)=\frac{40x-4x^2}{6}\]
|
(c)
|
Maximum value occurs at the turning point of the parabola. \[\therefore x=5\] \[\therefore A(5)=\frac{50}{3}\ m^2\]
|
(d)
|
Let \(k\) be the length of fencing required. \[\therefore A(x)=x\times \frac{k-4x}{6}\] \[\therefore x_{max}=\frac{k}{8}\] \[\therefore A(\frac{k}{8})=\frac{k^2}{96}\] \[\therefore \frac{k^2}{96}=150\] Therefore, 120 metres of fencing is required.
|
-
Success Criteria
-
Topic Resources
-
Practice Tests
-
VCAA Questions
-
Other Resources
<
>
Quadratic algebra:
- To be able to expand quadratic expressions including perfect squares of difference of perfect squares (DOPS)
- To be able to factorise quadratic expressions by removing a highest common factor.
- To be able to factorise quadratic trinomials by inspection "easy" (monic) and by-inspection "hard" (non-monic).
- To be able to factorise a quadratic expression by completing the square and using DOPS.
- To be able to understand and apply the null factor law to solve quadratic equations (factorising first if necessary).
- To be able to complete the square on a quadratic expression and express the quadratic in turning point form.
- To be able to solve quadratic equations by first completing the square and then "undoing" the equation.
- To be able to use the general quadratic formula to solve quadratic equations.
- To be able to understand and use the discriminant to determine the number of solutions to a quadratic equation.
- To be able to determine if the solution(s) to a quadratic equation will be rational or irrational using the discriminant.
- To be able to sketch the graph of a quadratic function from general form, turning point for or factorised form.
- To be able to used algebraic skills to determine the coordinates of key features of quadratic graphs.
- To be able to solve simultaneous equations involving equations and hence state points of intersection.
- To be able to solving quadratic inequations, using a graph to assist where necessary.
- To be able to represent a family of quadratic graphs sharing common feature(s) using parameter(s).
- To be able to determine the equation for a quadratic by first selecting an appropriate quadratic rule.
- To be able to model real-world phenomena with quadratic functions.
- To be able to features of a problem from a quadratic model and/or graph.
- To be able to optimise practical situations involving quadratic functions.
- To be able to determine the average rate of change between two points on a parabola.
- To be able to find the instantaneous rate of change using a tangent to a parabola with the assistance of technology.
Quadratic Skills - Worksheet A
|
Quadratic Functions [Yr10]:
This playlist contains additional examples of quadratic algebra from a Year 10 level:
|
|