Come up with a new linear function that has a slope that falls in the range
−< < 1 0 m . Choose two different initial values. For this new linear function,
what happens to the function’s values after many iterations? Are the
function’s values getting close to a particular number in each case?
I have no idea where to even begin for this question, can someone help me?
no idea what
−< < 1 0 m
means.
Sure! I can help you with that. To come up with a linear function that has a slope falling in the range of -10 < m < 1, you need to choose two different initial values and determine how the function's values change after many iterations.
Here's a step-by-step guide on how to proceed:
1. Choose a slope value, m, such that -10 < m < 1. For example, let's take m = 0.5. This satisfies the given condition.
2. Choose two different initial values, let's say x₁ = 1 and x₂ = -2. These values will help us observe how the function's values change after many iterations.
3. Write the linear function in the form y = mx + b, where m is the slope and b is the y-intercept. Since we are only interested in the slope, you can ignore the y-intercept for now. So, the linear function becomes y = 0.5x.
4. For each initial value, plug it into the linear function to find the corresponding y-value:
For x₁ = 1, y₁ = 0.5(1) = 0.5
For x₂ = -2, y₂ = 0.5(-2) = -1
5. Repeat the process of plugging the resulting y-values back into the linear function for many iterations. Let's see what happens after 10 iterations:
y₁ = 0.5(0.5) = 0.25
y₂ = 0.5(-1) = -0.5
y₃ = 0.5(0.25) = 0.125
y₄ = 0.5(-0.5) = -0.25
y₅ = 0.5(0.125) = 0.0625
y₆ = 0.5(-0.25) = -0.125
y₇ = 0.5(0.0625) = 0.03125
y₈ = 0.5(-0.125) = -0.0625
y₉ = 0.5(0.03125) = 0.015625
y₁₀ = 0.5(-0.0625) = -0.03125
6. As you can see, the values of the function begin to oscillate closer to zero but never actually reach it. In each case, the function's values approach a particular number, which in this case is zero.
Therefore, for this linear function with a slope falling in the range -10 < m < 1, the function's values after many iterations get close to zero.