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 function’s values after many iterations? Are the
function’s values getting close to a particular number in each case?
(The linear function before this was f(4)=1/2*4+1=3 idk if this will help)
Can someone please explain what they are asking or give me an example of what they're asking for?
You posted this earlier and this repost is just as confusing as the first one.
from your f(4) = 1/2 * 4 + 1
I could guess that your function was f(x) = (1/2)x + 1
so I can change the slope to -1/2 which is -1 ≤ m ≤ 0, but now what ?
linear functions, unless they are vertical or horizontal, do not approach
any constant value.
This actually helped me out a lot! Thank you so much
Certainly! The question is asking you to create a linear function (in the form f(x) = mx + b) where the slope, m, is a negative number that falls between -1 and 0. Additionally, the question asks you to choose two different initial values for x.
For example, let's say we choose m = -0.5 and our initial values for x are x = 0 and x = 5. Plugging these values into the linear function, we get:
f(0) = -0.5(0) + b = b
f(5) = -0.5(5) + b = -2.5 + b
Now, the question asks what happens to the function's values after many iterations. To answer this, we need to consider what happens when we plug in larger and larger values of x into our function. The slope of the linear function, being negative and less than 0, means that as x increases, the function's values will decrease at a decreasing rate.
In this case, as x approaches infinity, the function's values will approach a particular number, which is its y-intercept, represented by b in the equation. So, in each case, the function's values will get closer and closer to the y-intercept value, regardless of the chosen initial values of x.
Please note that this general behavior holds true for any linear function with a negative slope between -1 and 0, regardless of the chosen values for m and the initial values for x. The specific numbers chosen for m and the initial values will affect the exact values the function takes on, but the overall behavior will remain the same.
Sure! The question is asking you to come up with a new linear function that has a slope between -1 and 0. In other words, the line should have a negative slope that is less than -1.
To create such a linear function, let's say we choose the slope m to be -0.5. The general form of a linear function is y = mx + b, where m is the slope and b is the y-intercept.
Let's choose two different values for the y-intercept (b) to create two different linear functions:
1) Linear function 1: Let's say we choose b = 2. The linear function equation becomes y = -0.5x + 2.
2) Linear function 2: Let's say we choose b = 5. The linear function equation becomes y = -0.5x + 5.
Now, to determine what happens to the function's values after many iterations, you can choose different values of x and find the corresponding y values to see if they converge to a particular number.
For example, let's plug in different values of x:
For Linear function 1:
- When x = 0, y = 2
- When x = 1, y = 1.5
- When x = 2, y = 1
- When x = 3, y = 0.5
- When x = 4, y = 0
- When x = 5, y = -0.5
- When x = 6, y = -1
- ....
You can keep calculating more pairs of x and y values, and you'll notice that as x increases or decreases, the y-values approach negative infinity. In other words, the function's values are getting closer to negative infinity.
Similarly, you can do the same calculations for Linear function 2 and observe that the y-values also approach negative infinity.
Therefore, in both cases, the function's values are getting close to negative infinity as x increases or decreases.