Come up with a new linear function that has a slope that falls in the range -1<m<0. Choose two different initial values. For this new linear function what happens to the functions values after many iterations? Are the functions values getting closer to a particular number in each case? I am having trouble with this question on my project. Someone please help. The two linear equations I choose were y=-0.5+1 and y=-0.4+2

Answer: The value function of the function decreases and the values are not getting close to a particular number since they will be decreasing indefinitely

To come up with a new linear function that has a slope in the range -1 < m < 0, we can choose any value for the y-intercept (or initial value) and the slope. Let's consider the two linear equations you provided:

1. y = -0.5x + 1
2. y = -0.4x + 2

For these linear functions, let's examine what happens to the function values after many iterations, and whether they approach a particular number.

If we substitute different values of x into each equation, we can find corresponding y values. Let's choose a range of x-values and calculate the y-values for each function:

For equation 1 (y = -0.5x + 1):
- If x = 0, y = -0.5(0) + 1 = 1
- If x = 1, y = -0.5(1) + 1 = 0.5
- If x = 2, y = -0.5(2) + 1 = 0
- If x = 3, y = -0.5(3) + 1 = -0.5

For equation 2 (y = -0.4x + 2):
- If x = 0, y = -0.4(0) + 2 = 2
- If x = 1, y = -0.4(1) + 2 = 1.6
- If x = 2, y = -0.4(2) + 2 = 1.2
- If x = 3, y = -0.4(3) + 2 = 0.8

After calculating these values, we can see that the y-values change as the x-values change. However, there is no specific value that the functions approach after many iterations.

The function values for each linear equation do not seem to converge to a particular number in each case. The y-values change as the x-values change, indicating that the functions do not approach a constant value over time.

It's important to note that the behavior of linear functions may vary depending on the initial values chosen for both the slope and y-intercept. The selected values in this case did not result in a converging behavior.

To come up with a new linear function that has a slope within the range -1 < m < 0, we can use the general form of a linear equation: y = mx + b, where m is the slope and b is the y-intercept.

Let's create a new linear function: y = -0.75x + 2.

Now, let's examine what happens to the function's values after many iterations. We can do this by plugging in different x-values and observing the corresponding y-values.

Let's choose two different initial values for x: x = 1 and x = 3.

For x = 1, substituting it into the equation gives us:
y = -0.75(1) + 2
y = -0.75 + 2
y = 1.25

The corresponding y-value for x = 1 is 1.25.

For x = 3, substituting it into the equation gives us:
y = -0.75(3) + 2
y = -2.25 + 2
y = -0.25

The corresponding y-value for x = 3 is -0.25.

By repeating this process for various x-values, we can generate a set of (x, y) points. Plotting these points on a graph will give us a straight line with a slope between -1 and 0.

Regarding whether the function values are getting closer to a particular number in each case, we need to take into account the nature of the linear function. In this case, since the slope is within the range -1 < m < 0, the function will decrease as x increases. Therefore, the function values will not converge towards a particular number, but they will continuously decrease as we move along the x-axis.