Graphic function

Graphic Function

Many functions are given as equations. In such cases, drawing the graph of the equation means obtaining the visual representation of the function defined by that equation.

You can draw the graph of a function if you know the limits of its independent variables (domain) and dependent variables (range).

Example 1

Draw the graphs of the following functions:

1. f(x) = $3x - 1$
2. g(x) = $x^2 - 2x - 1$
3. h(x) = $x^3$

Solution:

For f(x) = $3x - 1$:

This is a linear function. The domain and range are all real numbers.

Create a table of values:

x	f(x) = 3x - 1
-2	-7
-1	-4
0	-1
1	2
2	5

For g(x) = $x^2 - 2x - 1$:

This is a quadratic function.

Create a table of values:

x	g(x) = x² - 2x - 1
-2	5
-1	2
0	-1
1	-2
2	-1
3	2
4	5

For h(x) = $x^3$:

This is a cubic function.

Create a table of values:

x	h(x) = x³
-2	-8
-1	-1
0	0
1	1
2	8

Conclusion:

• The first graph is for a linear function.
• The second graph is for a quadratic function.
• The third graph is for a cubic function.

Example 2

Draw the graph of the function:

f(x) = $-1 + 6x - x^2$

Solution:

This is a quadratic function (a parabola).

Create a table of values:

x	f(x) = -1 + 6x - x²
-2	-21
-1	-8
0	-1
1	4
2	7
3	8
4	7
5	4
6	-1

Functions with More Than One Part

Some functions are made up of more than one part. When drawing their graphs, draw each part separately.

If the graph includes an endpoint:

• Use a solid dot if the endpoint is included.
• Use a hollow dot if the endpoint is not included.

Example 3

Sketch the graph of the following:

1. f(x) = x + 1 for x > 0
2. f(x) = x + 1 for x ≤ 0

Solution:

• For part (a), the graph is a straight line starting from values just greater than 0. Use a hollow dot at x = 0 since it is not included.
• For part (b), the graph is the same straight line but going left from x = 0. Use a solid dot at x = 0 because the endpoint is included.

Step Functions

A step function is a piecewise function that increases or decreases in jumps, rather than changing smoothly. Its graph looks like steps of a staircase, which is where the name comes from.

Characteristics of step functions:

1. The function has constant values within specific intervals of the domain.
2. The graph is made of horizontal line segments.
3. Each segment has an endpoint, which may be solid (●) or hollow (○):
   • A solid dot (●) means the endpoint is included (e.g. x ≤ a).
   • A hollow dot (○) means the endpoint is not included (e.g. x < a).

Example of a step function

Let

f(x) =

• 1 for 0 ≤ x < 2
• 2 for 2 ≤ x < 4
• 3 for 4 ≤ x ≤ 6

To draw the graph:

1. From x = 0 to x = 2, draw a horizontal line at y = 1. Use ● at x = 0 and ○ at x = 2.
2. From x = 2 to x = 4, draw a horizontal line at y = 2. Use ● at x = 2 and ○ at x = 4.
3. From x = 4 to x = 6, draw a horizontal line at y = 3. Use ● at x = 4 and ● at x = 6 (because x = 6 is included).

Real-life example

A taxi fare that charges:

• 5,000 Tsh for 0–5 km
• 7,000 Tsh for 6–10 km
• 9,000 Tsh for 11–15 km

This situation can be modeled by a step function because the fare changes suddenly after each range, not gradually.

Important notes:

• Step functions are also called discrete functions because they jump from one value to another.
• The most common example is the greatest integer function, written as f(x) = ⌊x⌋, which gives the greatest whole number less than or equal to x.

Example 5

Draw the graph of a step function.

Solution:

In step functions, the graph jumps from one value to another and looks like a staircase. Each "step" represents a constant value over an interval. These types of functions are called step functions.