## Quadratic Functions and Equations in One Variable

 1.1 Quadratic Functions and Equations

Quadratic Expression in One Variable
 Definition
• A quadratic expression in one variable is an algebraic expression that has the highest power variable is two.
• The basic form of a quadratic expression is $$ax^2 + bx + c$$, which is $$a, b\, \text{and} \,c$$  is a constant and $$a ≠ 0$$, $$x$$ is a variable.
• $$a$$ is the coefficient of $$x^2$$$$b$$ is the coefficient of $$x$$ and $$c$$ is a constant.
 Examples
$$x^2+5x-1\\-y^2+3y\\2m^2+7$$
 Tips
Besides $$x$$, other letters can be used to represents variables.

Relationship Between a Quadratic Function and Many-To-One Relation
 Quadratic Function, $$f(x)= ax^2+bx+c$$
• All quadratic functions have the same image for two different images.
• Many-to-one relation.
• Have two shapes of graph.
 Shapes of the Graph,  $$f(x)= ax^2+bx+c , a \neq0$$

• For the graph $$a<0$$$$(x_1,y_1)$$ is known as maximum point.
• For the graph $$a>0$$$$(x_2, y_2)$$ is known as minimum point.
 Tips
The curved shape of the graph of a quadratic function is called a parabola.
 Axis of Symmetry of The Graph of a Quadratic Function
• Definition: A straight line that is parallel to the $$y-$$axis and divides the graph into two parts of the same size and shape.
• The axis of symmetry will pass through the maximum and minimum point of the graph of the function as shown in the diagram above.
• Equation axis of symmetry , $$x= - \dfrac{b}{2a}$$.
 Effects of Changing the Values of $$a$$, $$b$$ and $$c$$ on Graphs of Quadratic Functions,  $$f(x)= ax^2 +bx +c$$ The value of $$a$$ determines the shape of the graph. The value of $$b$$ determines the position of the axis of symmetry.  The value of $$c$$ determines the position of the $$y-$$intercept.
 Forming a Quadratic Equation Based on a Situation A quadratic function is written in the form of $$f(x)= ax^2 +bx +c$$ while a quadratic equation is written in the general form  $$ax^2 +bx +c = 0$$.
Roots of a Quadratic Equation
 Definition
The root of a quadratic equation $$ax^2 +bx +c = 0$$ are the values of the variables, $$x$$ which satisfy the equation.
 Relationship Between the Roots of a Quadratic Equation and The Positions of the Roots
The roots of equation $$ax^2 +bx +c = 0$$ are the points of intersection of the graph of the quadratic functions $$f(x)= ax^2 +bx +c$$ and the $$x-$$axis which are also knowns as the $$x-$$intercepts.
Determine The Roots of a Quadratic Equation by:
 Factorisation Method
• A quadratic equation needs to be written in the form of $$ax^2 +bx +c = 0$$ before we carry out factorisation.
• Example: Determine the roots of this quadratic equations by using factorisation method $$x^2 - 5x + 6 = 0$$.
• Solution:
$$\,\,\,\,\,\,\,x^2-5x+6=0\\(x-3)(x-2)=0\\\,\,\,\,\,\,\,\,\,\,\,x=3\,\text{or }x=2$$
 Graphical Method
The roots of a quadratic equation $$ax^2 +bx +c = 0$$ can be obtained by using a graphical method by reading the values of $$x$$ which are the points of intersections of the graph.

## Quadratic Functions and Equations in One Variable

 1.1 Quadratic Functions and Equations

Quadratic Expression in One Variable
 Definition
• A quadratic expression in one variable is an algebraic expression that has the highest power variable is two.
• The basic form of a quadratic expression is $$ax^2 + bx + c$$, which is $$a, b\, \text{and} \,c$$  is a constant and $$a ≠ 0$$, $$x$$ is a variable.
• $$a$$ is the coefficient of $$x^2$$$$b$$ is the coefficient of $$x$$ and $$c$$ is a constant.
 Examples
$$x^2+5x-1\\-y^2+3y\\2m^2+7$$
 Tips
Besides $$x$$, other letters can be used to represents variables.

Relationship Between a Quadratic Function and Many-To-One Relation
 Quadratic Function, $$f(x)= ax^2+bx+c$$
• All quadratic functions have the same image for two different images.
• Many-to-one relation.
• Have two shapes of graph.
 Shapes of the Graph,  $$f(x)= ax^2+bx+c , a \neq0$$

• For the graph $$a<0$$$$(x_1,y_1)$$ is known as maximum point.
• For the graph $$a>0$$$$(x_2, y_2)$$ is known as minimum point.
 Tips
The curved shape of the graph of a quadratic function is called a parabola.
 Axis of Symmetry of The Graph of a Quadratic Function
• Definition: A straight line that is parallel to the $$y-$$axis and divides the graph into two parts of the same size and shape.
• The axis of symmetry will pass through the maximum and minimum point of the graph of the function as shown in the diagram above.
• Equation axis of symmetry , $$x= - \dfrac{b}{2a}$$.
 Effects of Changing the Values of $$a$$, $$b$$ and $$c$$ on Graphs of Quadratic Functions,  $$f(x)= ax^2 +bx +c$$ The value of $$a$$ determines the shape of the graph. The value of $$b$$ determines the position of the axis of symmetry.  The value of $$c$$ determines the position of the $$y-$$intercept.
 Forming a Quadratic Equation Based on a Situation A quadratic function is written in the form of $$f(x)= ax^2 +bx +c$$ while a quadratic equation is written in the general form  $$ax^2 +bx +c = 0$$.
Roots of a Quadratic Equation
 Definition
The root of a quadratic equation $$ax^2 +bx +c = 0$$ are the values of the variables, $$x$$ which satisfy the equation.
 Relationship Between the Roots of a Quadratic Equation and The Positions of the Roots
The roots of equation $$ax^2 +bx +c = 0$$ are the points of intersection of the graph of the quadratic functions $$f(x)= ax^2 +bx +c$$ and the $$x-$$axis which are also knowns as the $$x-$$intercepts.
Determine The Roots of a Quadratic Equation by:
 Factorisation Method
• A quadratic equation needs to be written in the form of $$ax^2 +bx +c = 0$$ before we carry out factorisation.
• Example: Determine the roots of this quadratic equations by using factorisation method $$x^2 - 5x + 6 = 0$$.
• Solution:
$$\,\,\,\,\,\,\,x^2-5x+6=0\\(x-3)(x-2)=0\\\,\,\,\,\,\,\,\,\,\,\,x=3\,\text{or }x=2$$
 Graphical Method
The roots of a quadratic equation $$ax^2 +bx +c = 0$$ can be obtained by using a graphical method by reading the values of $$x$$ which are the points of intersections of the graph.