Higher Mathematics · Classes 9-10 · Chapter 1

Graphs of Relations and Functions.

Put a function on paper and a lot shows at a glance: straight or curved, where it bottoms out, and whether it is a function at all. The powers in the equation already tell you what shape to expect.

1. How a graph is drawn

A graph is the geometric picture of a function. Two lines XOX′ and YOY′ are drawn at right angles through a point O. O is the origin, XOX′ the x-axis and YOY′ the y-axis.

To draw y = f(x), make a table of values of the independent variable x and the dependent variable y on some interval a ≤ x ≤ b. Plot the points on the xy-plane, then join them with a straight line or a smooth curve.

2. Linear functions

The general form of a linear function is f(x) = mx + b, where m and b are real numbers. Its graph is a straight line with slope m and y-intercept b. The same line is often written y = mx + c, and then c is the y-intercept.

With m = 3 and b = 2, f(x) = 3x + 2. For x = −2, −1, 0, 1, 2 we get y = −4, −1, 2, 5, 8. The points (−2, −4), (−1, −1), (0, 2), (1, 5), (2, 8) all lie on one straight line. Change m and b and watch the line turn and slide.

If the equation comes in another shape, first write y as a function of x. y − 2 = 3(x − 5) ⟹ y = 3x − 13; y − 5 = −2(x + 1) ⟹ y = −2x + 3; x + y = 3 ⟹ y = 3 − x; 2x − y + 5 = 0 ⟹ y = 2x + 5; 2x − y + 6 = 0 ⟹ y = 2x + 6. Every one is a straight line, and every one is a function.

3. Quadratic functions

A quadratic function is described by y = ax² + bx + c, where a, b, c are real numbers and a ≠ 0. With a = 0 the x² term disappears and the function is linear.

Take a = 1, b = −4, c = −1: y = x² − 4x − 1. For x = −1, 0, 1, 2, 3, 4, 5, y = 4, −1, −4, −5, −4, −1, 4. The lowest point is (2, −5), and x = 1 and 3, 0 and 4, −1 and 5 give the same y. Tap each row of the table below to work out its y.

The table already shows the three properties of a quadratic function. Its graph is a parabola. It is symmetric about the y-axis or a line parallel to it. And the function takes its least or greatest value at one point. So the graphs of y = x², y = x² + 4x + 1 and x² + y = 9 are parabolas too.

4. Circles

If p, q, r are constants and r ≠ 0, the graph in R of the relation S = {(x, y) : (x − p)² + (y − q)² = r²} is a circle with centre (p, q) and radius r. On graph paper it is drawn with a compass, centred at (p, q), with radius r.

Example 35: the graph of S = {(x, y) : x² + y² = 16} is the circle x² + y² = 4², centre (0, 0), radius 4. It passes through (4, 0), (0, 4), (−4, 0) and (0, −4). In the same way x² + y² = 9 has centre (0, 0) and radius 3, and x² + y² = 25 has radius 5.

When the centre is not the origin, complete the squares to reach the circle's form. x² + y² + 2x − 6 = 0 ⟹ (x + 1)² + y² = 7, centre (−1, 0). x² + y² + 4x + 3y = 0 ⟹ (x + 2)² + (y + 3/2)² = 25/4, centre (−2, −3/2).

5. Function or not, from the graph

A function gives each x exactly one y. On a graph that means: if some vertical line, parallel to the y-axis, cuts the graph at more than one point, one x has two y values and the relation is not a function. If every vertical line cuts it at most once, it is a function.

So the circle x² + y² = 25 is not a function: at x = 3, both y = 4 and y = −4. Add the condition y ≥ 0 and only the upper half is left (0 ≤ y ≤ 5); with y ≤ 0, the lower half (−5 ≤ y ≤ 0): both are functions. But x ≥ 0 gives the right half-circle, which reaches both above and below the x-axis: that is not a function.

The graph of x = −2 is a line parallel to the y-axis: x = −2 has infinitely many y, so it is not a function. The graph of y = −2 is parallel to the x-axis: every x has a single y, so it is a function, though not a one-one function.

y = x² is a parabola opening upwards, and a function. But the graph of y² = x is a parabola opening to the right: the same x gives y = ±√x, two values, so it is not a function.

Pick a graph, then slide the vertical line x = c left and right. Count how many times it cuts the graph at each position.

Nine graphs, one at a time. Say whether each one is a function.

6. Check yourself

In the first, look at the powers.

In the second, complete the squares.

For the last, picture a vertical line.

The whole note on eight cards.

Source: Higher Mathematics, Classes 9-10 (NCTB), Chapter 1 “Set and Function”, the “Graphs of Relations and Functions” part.