KS5 Grade 11 - QUADRATIC FUNCTIONS AND EQUATIONS
Lesson 6...APPLICATIONS OF QUADRATICS
![Picture](/uploads/1/3/3/4/13343229/2754331.jpg)
Quadratic functions and their graphs can be used to model many different situations (Eg. the flight path of a thrown object). Using the knowledge and methods covered in the previous lessons solving any problems involving quadratics should now be possible
TASK 1 - Textbook Pg55 Ex2K
HOMEWORK - Textbook Pg56/57 Review Ex's
TASK 1 - Textbook Pg55 Ex2K
HOMEWORK - Textbook Pg56/57 Review Ex's
Lesson 5...GRAPHS OF QUADRATIC FUNCTIONS (2)
![Picture](/uploads/1/3/3/4/13343229/3262948.gif)
When shown the graph of a quadratic function it is necessary to be able to use the information given on the graph to findi the equation of the function
Through looking at the shape of the graph and then identifying any y-intercepts, x-intercepts, co-ordinates of the vertex or the axis of symmetry, it is possible to write work out the equation in any of the 3 forms (Standard, Turning Point, Factorised)
TASK 1 - Textbook Pg52 Ex2J
HOMEWORK - None set
Through looking at the shape of the graph and then identifying any y-intercepts, x-intercepts, co-ordinates of the vertex or the axis of symmetry, it is possible to write work out the equation in any of the 3 forms (Standard, Turning Point, Factorised)
TASK 1 - Textbook Pg52 Ex2J
HOMEWORK - None set
Lesson 4...GRAPHS OF QUADRATIC FUNCTIONS (1)
![Picture](/uploads/1/3/3/4/13343229/3503609.jpg?283)
Graphs of quadratic functions have a curved shape known as a PARABOLA
The VERTEX of a graph of a quadratic function is the minimum or maximum point
Graphs of quadratic functions are symmetrical about an AXIS OF SYMMETRY which runs vertically through the vertex of the parabola
For quadratic functions in standard form y = ax^2 + bx + c, the graph crosses the y-axis (y-intercept) at (0, c) and the equation of the axis of symmetry is x = - b/2a
When the basic quadratic function of y = x^2 undergoes transformations, the resulting functions can be written (in turning-point form) as y = a(x - h)^2 + k. Examples of this can be viewed in the diagram above. When a quadratic function is in the form previously mentioned, the graph has its vertex at (h, k)
TASK 1 - Textbook Pg46 Ex2H
HOMEWORK - Read Textbook Pg47 and attempt Pg48 Ex 2I
The VERTEX of a graph of a quadratic function is the minimum or maximum point
Graphs of quadratic functions are symmetrical about an AXIS OF SYMMETRY which runs vertically through the vertex of the parabola
For quadratic functions in standard form y = ax^2 + bx + c, the graph crosses the y-axis (y-intercept) at (0, c) and the equation of the axis of symmetry is x = - b/2a
When the basic quadratic function of y = x^2 undergoes transformations, the resulting functions can be written (in turning-point form) as y = a(x - h)^2 + k. Examples of this can be viewed in the diagram above. When a quadratic function is in the form previously mentioned, the graph has its vertex at (h, k)
TASK 1 - Textbook Pg46 Ex2H
HOMEWORK - Read Textbook Pg47 and attempt Pg48 Ex 2I
Lesson 3...ROOTS OF QUADRATIC EQUATIONS
![Picture](/uploads/1/3/3/4/13343229/9526425.jpg)
Using the methods of factorising, completing the square or solving by the quadratic formula, we are shown the roots of a quadratic equation. More specifically these are any x-intercept values for the graph of the quadratic.
The discriminant, Delta (Δ), can give us information about the roots of an equation as follows
If b^2 - 4ac > 0, the equation will have 2 different real roots
If b^2 - 4ac = 0, the equation will have only one solution
If b^2 - 4ac < 0, the equation will have no real roots
This can be represented graphically as seen in the image on the left
TASK 1 - Textbook Pg42 Ex2G
Homework - Finish TASK 1
The discriminant, Delta (Δ), can give us information about the roots of an equation as follows
If b^2 - 4ac > 0, the equation will have 2 different real roots
If b^2 - 4ac = 0, the equation will have only one solution
If b^2 - 4ac < 0, the equation will have no real roots
This can be represented graphically as seen in the image on the left
TASK 1 - Textbook Pg42 Ex2G
Homework - Finish TASK 1
Lesson 2...SOLVING QUADRATIC EQUATIONS (2)
![Picture](/uploads/1/3/3/4/13343229/631397.jpg)
Solving quadratics by using the QUADRATIC FORMULA
For quadratic equations that are to hard to factorise, the quadratic formula (see image) can be used
where a, b and c are defined as follows
ax^2 + bx + c = 0
TASK 1 - Textbook Pg40 Ex2E/2F
Homework - None set
For quadratic equations that are to hard to factorise, the quadratic formula (see image) can be used
where a, b and c are defined as follows
ax^2 + bx + c = 0
TASK 1 - Textbook Pg40 Ex2E/2F
Homework - None set
Lesson 1...SOLVING QUADRATIC EQUATIONS (1)
![Picture](/uploads/1/3/3/4/13343229/5945061.jpg?285)
A quadratic equation is one that can be written or rearranged into the form
ax^2 + bx + c = 0 where a ≠ 0, ax^2 is the quadratic term, bx is the linear term and c is the constant term
Solving quadratics by FACTORISATION
By factorising a quadratic into a pair of brackets and then by using the following property...
If (x - a)(x - b) = 0, then either x - a = 0 or x - b = 0
...a simple quadratic equation can be solved as in the following example
Eg. x^2 - 5x - 14 = 0 gives
(x - 7)(x + 2) = 0
Therefore, x - 7 = 0 or x + 2 = 0
x = 7 or x = - 2
Solving quadratics by COMPLETING THE SQUARE
To create, and solve, a quadratic equation consisting of a perfect square trinomial, we use a method called completing the square as follows
Ensure the coefficient of x^2 is 1, otherwise factor out the coefficient or divide through by it
Take half the coefficient of x
Square it, and add the result to both sides of the equation
Now rearrange the equation to solve for x (see full example in image above)
TASK 1 - Textbook Pg35 Ex2A/2B
TASK 2 - Textbook Pg37/38 Ex2C/2D
HOMEWORK - Finish TASK 1 and TASK 2
ax^2 + bx + c = 0 where a ≠ 0, ax^2 is the quadratic term, bx is the linear term and c is the constant term
Solving quadratics by FACTORISATION
By factorising a quadratic into a pair of brackets and then by using the following property...
If (x - a)(x - b) = 0, then either x - a = 0 or x - b = 0
...a simple quadratic equation can be solved as in the following example
Eg. x^2 - 5x - 14 = 0 gives
(x - 7)(x + 2) = 0
Therefore, x - 7 = 0 or x + 2 = 0
x = 7 or x = - 2
Solving quadratics by COMPLETING THE SQUARE
To create, and solve, a quadratic equation consisting of a perfect square trinomial, we use a method called completing the square as follows
Ensure the coefficient of x^2 is 1, otherwise factor out the coefficient or divide through by it
Take half the coefficient of x
Square it, and add the result to both sides of the equation
Now rearrange the equation to solve for x (see full example in image above)
TASK 1 - Textbook Pg35 Ex2A/2B
TASK 2 - Textbook Pg37/38 Ex2C/2D
HOMEWORK - Finish TASK 1 and TASK 2