Challenge: 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.
y = - (x + 1)^2 - 4
x y
-3
-2
-1
0
1
To determine if the function is linear or nonlinear, we can start by creating a table and calculating the values of y for different x values.
x y
-3 -9
-2 -1
-1 -5
0 -5
1 -9
Now, let's plot these points on a graph.
Here is a rough sketch of the graph:
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-3 -2 -1 0 1
```
From both the table and the graph, we can see that the function does not have a constant rate of change. This indicates that the function is nonlinear.
Now let's find a linear function. A linear function has a constant rate of change, which means that the difference between any two y-values is the same for any two respective x-values.
To find a linear function, we need two points. Let's use the points (-2, -1) and (1, -9) from the table and graph.
The equation for a linear function can be written as y = mx + b, where m is the slope and b is the y-intercept.
First, let's find the slope (m) using the formula:
m = (y2 - y1) / (x2 - x1)
m = (-9 - (-1)) / (1 - (-2))
m = (-9 + 1) / (1 + 2)
m = -8 / 3
Now, let's find the y-intercept (b) using the formula:
b = y - mx
b = -1 - (-8/3)(-2)
b = -1 - (16/3)
b = -1 - (16/3)
b ≈ -1 + 5.33
b ≈ 4.33
Therefore, the linear function that fits the given data is:
y = (-8/3)x + 4.33
By comparing the table and graph of the original function with this linear function, we can see that the original function is nonlinear because it does not have a constant rate of change.