1st Year Math Chapter 2 Exercise 2.2 Notes: Graphs of Functions (Punjab Board 2026-27)
Complete concept notes, formulas, and solved examples for ICS/FSc Part 1 Mathematics, Punjab Board (PECTAA), Single National Curriculum 2026-27 session.
Exercise 2.2 builds on the function concepts from Exercise 2.1 and focuses on graphing functions — plotting linear and quadratic functions, finding intercepts, and reading a function’s domain and range directly from its graph.
What Does This Exercise Cover?
This exercise connects the algebraic rule of a function to its visual representation, teaching how slope, intercepts, and shape reveal a function’s behavior at a glance.
Key Concepts
1. Graphing a Function
To graph a function, calculate f(x) for several values of x, plot the resulting points (x, f(x)) on a coordinate plane, and connect them to reveal the function’s shape.
2. Linear Functions
A linear function has the form f(x) = mx + c, and its graph is always a straight line. Here, m is the slope (how steeply the line rises or falls) and c is the y-intercept (where the line crosses the y-axis).
3. Quadratic Functions
A quadratic function has the form f(x) = ax² + bx + c, and its graph is a curve called a parabola. If a > 0, the parabola opens upward; if a < 0, it opens downward.
4. Intercepts
The y-intercept is found by evaluating f(0). The x-intercept(s) are found by setting f(x) = 0 and solving for x — these are the points where the graph crosses each axis.
5. Reading Domain and Range from a Graph
The domain of a function can be read from the horizontal extent (left-right spread) of its graph, while the range can be read from the vertical extent (up-down spread).
Solved Examples
Example 1: Graph f(x) = 2x + 3, identifying its slope and y-intercept.
Comparing with f(x) = mx + c gives slope m = 2 and y-intercept c = 3.
At x = 0, f(0) = 3, giving the point (0, 3).
At x = −1, f(−1) = 2(−1) + 3 = 1, giving the point (−1, 1).
Plotting (0, 3) and (−1, 1) and drawing a line through them gives the graph.
Answer: Slope = 2, y-intercept = 3, line passes through (0, 3) and (−1, 1)
Example 2: Find the x-intercept and y-intercept of f(x) = x − 4.
y-intercept: evaluate f(0) = 0 − 4 = −4, giving the point (0, −4).
x-intercept: set f(x) = 0, so x − 4 = 0, giving x = 4, or the point (4, 0).
Answer: x-intercept = (4, 0), y-intercept = (0, −4)
Example 3: For f(x) = x², describe the direction the parabola opens and state its vertex.
Comparing with f(x) = ax² + bx + c gives a = 1, which is positive.
Since a > 0, the parabola opens upward.
There is no linear or constant term, so the lowest point (vertex) is at the origin, (0, 0).
Answer: The parabola opens upward, with vertex at (0, 0)
Sample MCQs
1. In f(x) = mx + c, the value m represents:
a) The y-intercept b) The slope c) The x-intercept d) The domain
Answer: b) The slope
2. The graph of f(x) = ax² + bx + c is a:
a) Straight line b) Parabola c) Circle d) Hyperbola
Answer: b) Parabola
3. The y-intercept of a function is found by setting:
a) y = 0 b) x = 0 c) x = y d) f(x) = x
Answer: b) x = 0
4. If a > 0 in f(x) = ax² + bx + c, the parabola opens:
a) Upward b) Downward c) Sideways d) It is a straight line
Answer: a) Upward
5. The x-intercept of f(x) = x − 5 is:
a) (0, −5) b) (5, 0) c) (0, 5) d) (−5, 0)
Answer: b) (5, 0)
Important Short Questions
- What does the slope m represent in the graph of f(x) = mx + c?
- How is the y-intercept of a function found algebraically?
- Explain how to find the x-intercept of a function.
- Describe the general shape of the graph of a quadratic function.
- Find the y-intercept of f(x) = 3x − 7.
Important Long Questions
- Graph f(x) = −2x + 4, labeling the slope and both intercepts.
- Find the x-intercept and y-intercept of f(x) = 2x² − 8, and describe the resulting graph.
- Explain how the domain and range of a function can be read directly from its graph.
- Compare the graphs of f(x) = x² and f(x) = −x², explaining the effect of the negative sign.
How to Approach This Exercise Effectively
- For any linear function, plot just two points — the y-intercept and one other point using the slope — and draw a straight line through them.
- Always find intercepts algebraically first (by setting x = 0 or f(x) = 0), then use them to guide your graph, rather than guessing points.
- Remember the sign of a in a quadratic function immediately tells you whether the parabola opens upward or downward — check this before graphing.
- Practice reading domain and range off both hand-drawn and printed graphs, not just from algebraic rules.
- Label every intercept and key point clearly on your graph — many marks in this exercise come from correctly labeled diagrams, not just the sketch itself.
FAQs
Q: What is the fastest way to graph a linear function?
A: Identify the y-intercept from the equation, plot it, then use the slope to find a second point by moving accordingly (rise over run), and draw a straight line through both.
Q: How many x-intercepts can a quadratic function have?
A: A quadratic function can have zero, one, or two x-intercepts, depending on whether its graph touches or crosses the x-axis.
Q: Why does the sign of a matter in a quadratic function?
A: It determines the direction the parabola opens: positive a opens upward (a minimum point), while negative a opens downward (a maximum point).
Q: Can I find the domain and range without graphing?
A: For simple functions like straight lines and parabolas, yes — but sketching the graph makes it far easier to see the domain and range at a glance, especially for more complex functions.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 2: Functions and Graphs, Exercise 2.2.