Quadratic Functions and Equations in One Variable

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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.