Special functions and their graphs
Know the shapes: identity and linear give straight lines, constant a flat line, modulus a V, quadratic a parabola, cubic an S-curve and reciprocal a hyperbola.
The special functions
| Function | Rule | Shape of the graph | Example |
|---|---|---|---|
| Identity | f(x) = x | Straight line through the origin, making 45° with the x-axis | f(5) = 5 |
| Constant | f(x) = c | Horizontal line y = c; range = {c} | f(x) = 4 for every x |
| Linear | f(x) = mx + c, m ≠ 0 | Straight line that is not horizontal or vertical | f(x) = 2x + 1 |
| Modulus (absolute value) | f(x) = |x|: x when x ≥ 0, and −x when x < 0 | V shape with its corner at the origin; never below the x-axis, so the range is [0, ∞) | f(−3) = |−3| = 3 |
| Quadratic | f(x) = ax² + bx + c, a ≠ 0 | Parabola; opens upward if a > 0, downward if a < 0 | f(x) = x² − 4 |
| Cubic | f(x) = ax³ + bx² + cx + d, a ≠ 0 | A curve like a stretched S; y = x³ runs from the third quadrant to the first | f(x) = x³ |
| Reciprocal | f(x) = 1/x, x ≠ 0 | Two curved branches in the first and third quadrants (a rectangular hyperbola); never meets the axes | f(2) = 1/2 |
Two more words from your textbook
A function f: A → B is a real valued function when its range is a subset of R, the set of real numbers. Every function in the table is real valued. The modulus function is not linear. Its graph is made of two straight pieces: y = x for x ≥ 0 and y = −x for x < 0.
How to draw the graph of y = x²
- 1Make a table for x = −3, −2, −1, 0, 1, 2, 3. The y values are 9, 4, 1, 0, 1, 4, 9.
- 2Draw the axes on graph paper. Take 1 cm = 1 unit on both axes and write the scale.
- 3Plot (−3, 9), (−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4) and (3, 9).
- 4Join the points with a smooth U-shaped curve, freehand, not with a ruler. Label it y = x².
- 5Note the shape: it is symmetric about the y-axis, and its lowest point (the vertex) is the origin.
Example 1: recognise the function
Name the type of each function: (i) f(x) = 7 (ii) g(x) = 3 − 2x (iii) h(x) = 2x² − x + 5 (iv) p(x) = x³ − 1 (v) q(x) = 1/x, x ≠ 0.
- (i) The output is always 7, so f is a constant function.
- (ii) The highest power of x is 1, so g is a linear function.
- (iii) The highest power of x is 2, so h is a quadratic function.
- (iv) The highest power of x is 3, so p is a cubic function.
- (v) x is in the denominator, so q is the reciprocal function.
Constant, linear, quadratic, cubic and reciprocal.
Example 2: a linear function in real life
A cab charges a fixed ₹ 50 plus ₹ 12 per km (example rates). Write the fare as a function of the distance x km. Find the fare for 15 km, and the distance travelled when the fare is ₹ 290.
- Fare f(x) = 12x + 50. It has the form mx + c with m = 12 ≠ 0, so it is linear.
- f(15) = 12 × 15 + 50 = 180 + 50 = 230.
- 12x + 50 = 290 ⇒ 12x = 240 ⇒ x = 20.
- Check: f(20) = 240 + 50 = 290.
f(x) = 12x + 50; the fare for 15 km is ₹ 230; ₹ 290 pays for 20 km.
Buying paneer at an example rate of ₹ 400 per kg gives a linear function: cost = 400 × weight. A fixed monthly fee that does not change with use is a constant function.
1 marks: “The graph of f(x) = 1/x is: (1) a straight line (2) a parabola (3) a rectangular hyperbola (4) a circle.” (3) a rectangular hyperbola.
2 marks: “f: R → R is defined by f(x) = 3x − 2. Name the type of function and find f(4) − f(1).” Linear, because it has the form mx + c with m = 3 ≠ 0. f(4) − f(1) = 10 − 1 = 9.
5 marks: “A tank holds 500 litres of water and drains at 20 litres per minute. (i) Write the volume V(t) left after t minutes. (ii) Name the type of function. (iii) Find V(10). (iv) When is the tank empty?” V(t) = 500 − 20t, a linear function. V(10) = 500 − 200 = 300 litres. 500 − 20t = 0 gives t = 25 minutes, so the domain is 0 ≤ t ≤ 25.
The highest power tells the shape: power 0 flat, 1 straight, 2 parabola, 3 S-curve; x in the denominator gives a hyperbola.