Question

Decide whether the function is linear or nonlinear using both a table and a graph. Use pencil and paper. Find a linear function. Make a table and a graph for the function to explain why it is linear

Answer 1

To determine whether a function is linear or nonlinear, we need to examine the behavior of the function and determine if it follow the properties of a linear function.

A linear function has a constant rate of change and forms a straight line when graphed. The equation of a linear function can be written in the form y = mx + b, where m is the slope and b is the y-intercept.

Let's consider a function given by a table of values:

Input (x) | Output (y)
--------------------
1 | 2
2 | 4
3 | 6
4 | 8

To determine if this function is linear, we need to check if the rate of change is constant. We can calculate the rate of change by finding the difference in the output (y) values for any two corresponding input (x) values.

For example, when x increases from 1 to 2, y increases from 2 to 4, resulting in a rate of change of (4 - 2)/(2 - 1) = 2. Similarly, for the other pairs, the rate of change is consistently 2. Therefore, the function has a constant rate of change, indicating linearity.

Next, let's graph this function using the given table:

(x-axis)
|
-----------------
|
|
|
|
|------------------
| (4,8)
|
|
|
|
|
|
------------------(1,2)
(y-axis)

The graph of the function forms a straight line passing through the points (1,2), (2,4), (3,6), and (4,8). This confirms linearity.

Now, let's find the equation of the linear function using the slope-intercept form y = mx + b. We can use any set of (x, y) coordinates from the table to find the slope (m), and then substitute the slope and one of the points into the equation to solve for the y-intercept (b).

Considering the points (1,2) and (2,4), the slope (m) can be calculated as (4 - 2)/(2 - 1) = 2.

Using the point (1,2) and the slope (2) in the equation y = mx + b, we have:
2 = 2(1) + b
2 = 2 + b
b = 2 - 2
b = 0

The equation of the linear function is y = 2x.

In summary, the given function is linear because it has a constant rate of change and forms a straight line when graphed. The equation of the linear function is y = 2x.