Class 9 Mathematics | Punjab Curriculum and Textbook Board Syllabus 2025
Graphs help us see and understand how different things are related to each other.
They are very useful in explaining mathematical functions and how these functions work in real-life situations.
A function represents a relationship between two variables, where one variable depends on the other. It can be expressed as an equation, a graph, a numerical table, or a verbal description.
A function is written as:
\( y = f(x) \)
Here:
A linear function shows a straight-line relationship between two variables. Its general form is:
\( f(x) = mx + c \)
or
\( y = mx + c \)
Where:
A quadratic function is a type of polynomial function that includes the term \( x^{2} \). The general form of a quadratic function is:
\( y = ax^{2} + bx + c \)
Where \( a \), \( b \), and \( c \) are constants and \( a \neq 0 \).
Note: The graph of a quadratic function is always a parabola.
A cubic function is a type of polynomial function of degree 3. Its standard form is:
\( y = ax^{3} + bx^{2} + cx + d \)
Where \( a \), \( b \), \( c \) and \( d \) are constants and \( a \neq 0 \).
Note:
A reciprocal function is a function of the form:
\( y = \frac{a}{x} \)
Where \( a \) is any real number and \( x \neq 0 \).
Note: An asymptote is a line that a graph approaches but never touches.
An exponential function is a mathematical function of the form:
\( y = ka^{x} \)
Where \( a \), \( k \) are constants, \( x \) is a variable, and \( a > 1 \).
The gradient or slope of a graph at any point is equal to the gradient of the tangent to the curve at that point. The gradient between two points is defined as:
\( \text{Gradient} = \frac{\text{Change in } y}{\text{Change in } x} = \frac{\Delta y}{\Delta x} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \)
Note: A tangent is a line that just touches a curve only at one point (and doesn't cross it).