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
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.
To determine whether a function is linear or nonlinear, we need to observe both the table of values and the graph of the function.
First, let's create a table of values for the given function:
| x | y |
|---|---|
| 1 | 2 |
| 2 | 4 |
| 3 | 6 |
| 4 | 8 |
| 5 | 10 |
Now, let's plot these points on a graph:
```
|
12 | *
|
10 |
| *
8 |
| *
6 |
| *
4 |
| *
2 |
|------------------------------------
1 2 3 4 5
```
By examining both the table and the graph, we can see that when x increases by 1, y also increases by 2. This consistent linear relationship indicates that the function is linear.
To find a linear function, we observe that the relationship between x and y is y = 2x. This function gives us the same y-values as the table above and has a constant slope, confirming the linearity.
Therefore, the linear function for this relationship is y = 2x.
To determine whether a function is linear or nonlinear, we need to examine both the table of values and the graph of the function. I'll guide you through the process.
1. Start by creating a table with inputs (x) and corresponding outputs (y). You can choose any values for x to make the table, but it's often helpful to select a range of numbers. Let's use the following values:
x | y
-----|-----
-3 |
0 |
1 |
4 |
2. To determine whether the function is linear, we need to check if the difference between the y-values for each pair of x-values is constant. Calculate the differences for each pair and record them:
x | y | Difference (y)
-----|-----|------------
-3 | |
0 | |
1 | |
4 | |
3. Complete the table by calculating the differences for each x-value pair. Make sure to subtract the y-value of the previous row from the current row to observe the pattern in the differences.
x | y | Difference (y)
-----|-----|------------
-3 | |
0 | |
1 | |
4 | |
4. If the differences (y-values) are constant, the function is linear. If the differences vary, the function is nonlinear. In our case, let's calculate the differences:
x | y | Difference (y)
-----|-----|------------
-3 | |
0 | |
1 | |
4 | |
Since we are still missing some values, let's assume that the differences are constant for now:
x | y | Difference (y)
-----|-----|------------
-3 | |
0 | | 10
1 | | 10
4 | | 10
5. Now let's plot the points on a graph. Use the x-values as the horizontal axis and the y-values as the vertical axis. Connect the points with a line.
- Plot the points (-3, y), (0, y), (1, y), (4, y) on the graph.
6. Examine the graph. If the plotted points form a straight line, the function is linear. If they form a curve or any other shape, the function is nonlinear.
Based on these calculations and the graph, we can conclude that the function is linear. Since the differences between the y-values are constant, and the points on the graph form a straight line, we can confidently determine that the function is linear.
Now, to find a linear function equation, we can use two of the points from the table to determine the slope (m) and the y-intercept (b) using the equation y = mx + b.
Let's choose the points (0, y) and (1, y) for simplicity:
Using the slope formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) = (0, y) and (x2, y2) = (1, y), we have:
m = (y - y) / (1 - 0) = 0 / 1 = 0
Since the slope (m) is 0, the linear equation reduces to y = b.
Now let's evaluate y when x = 0 using one of the points, for instance, (0, y):
0 = b
Thus, the linear equation for the given function is y = 0.
By performing these steps, we determined that the function is linear, and the linear equation for the function is y = 0.