Introduction
HSA Notes brings short, clear, exam focused notes designed to help Class 9 SSC Part One Matric students score 90 plus marks in Punjab Board exams under the New Book 2026 to 2027 session. This page covers Unit 10, Graphs of Functions, from Exercise 10.1, including recognising and sketching the graphs of linear and quadratic functions.
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HSA Notes Features
- Short, easy to remember definitions.
- Step by step solutions for every question.
- Tricks and memory aids for faster revision.
- Exact alignment with the PCTB and PECTAA textbook.
- Graphs simplified into a clear, repeatable method.
- Past paper pattern questions included throughout.
- A fast revision format built for last minute review.
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Available Across All Punjab Boards and Top Schools
These notes are built for students across all Punjab Boards, including Lahore, Gujranwala, Rawalpindi, Faisalabad, Bahawalpur, Multan, Sargodha, Sahiwal, and DG Khan, and are used at top schools including The City School, Beaconhouse, Roots International, Lahore Grammar School, Karachi Grammar School, Froebel’s, The Educators, Army Public School, Divisional Public School, and Government Model School.
Exercise 10.1, Graphs of Functions
| No. | Function | Type of Graph |
|---|---|---|
| i | y = 3x − 5 | Linear, a straight line |
| ii | y = −2x² + 3 | Quadratic, a parabola |
View the PDF for the full plotted diagrams of both graphs.
How to Recognise a Linear Function from Its Equation
A function is linear whenever x appears only to the first power, with no squared or higher terms, giving an equation of the form y = mx + c.
In this form, m is the slope of the line, telling you how steeply and in which direction the line rises or falls, and c is the y-intercept, the point where the line crosses the y-axis. For y = 3x − 5, the slope is 3, meaning the line rises 3 units for every 1 unit moved to the right, and the y-intercept is −5, so the line crosses the y-axis at the point (0, −5).
Because a linear function has a constant slope, its graph is always a single straight line, and only two points are needed to plot it accurately, though a third point is a useful check against arithmetic mistakes.
How to Recognise a Quadratic Function from Its Equation
A function is quadratic whenever the highest power of x is 2, giving an equation of the general form y = ax² + bx + c, and its graph is always a parabola, a smooth U-shaped or inverted U-shaped curve.
The sign of a, the coefficient of x², decides which way the parabola opens: if a is positive the parabola opens upward, and if a is negative it opens downward. For y = −2x² + 3, the coefficient of x² is −2, which is negative, so the parabola opens downward, with its highest point, the vertex, occurring where x = 0, giving the point (0, 3).
Plotting a quadratic accurately means calculating several y-values on either side of the vertex, since the curve is symmetric about the vertical line passing through it, which makes it easy to check that the two sides mirror each other correctly.
Why This Unit Matters for Your Overall Score
Graphs of functions connect the algebra learned throughout the year to a visual picture, and being able to identify a function’s type directly from its equation, before ever plotting a single point, saves valuable time in the exam room.
This skill also carries forward into later work on solving equations graphically and interpreting real-world situations modelled by linear or quadratic relationships, making a confident grasp of this unit useful well beyond its own exercises.
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Frequently Asked Questions
How do I know if a function’s graph will be a straight line? If x appears only to the first power in the equation, in the form y = mx + c, the graph is a straight line.
How do I know if a function’s graph will be a parabola? If the highest power of x in the equation is 2, in the form y = ax² + bx + c, the graph is a parabola.
How do I tell which way a parabola opens? Check the sign of the coefficient of x². A positive coefficient means the parabola opens upward, and a negative coefficient means it opens downward.
Are these notes aligned with the PCTB and PECTAA textbook? Yes, these notes are built to exactly match the PCTB and PECTAA New Book for Class 9 Mathematics for the 2026 to 2027 session.
Conclusion
With Exercise 10.1 fully solved, covering the recognition of linear and quadratic functions from their equations, Class 9 students have a strong foundation for Unit 10, Graphs of Functions. Use these notes alongside regular practice with past papers to build the confidence needed for 90 plus marks in your Punjab Board exam.
Best of luck for your Class 9 Mathematics exam!