Graph the recursive function below, complete the table of values, and rewrite it as an explicit function (slope-intercept or point slope) 푎 0 = − 6; 푎 푛 = 푎 푛−1 + 2.5
To graph the recursive function and complete the table of values, we can start with the given initial value of a_0 = -6 and use the recursive formula a_n = a_(n-1) + 2.5 to generate the subsequent terms.
Table of values:
n | a_n
------------
0 | -6
1 | -6 + 2.5 = -3.5
2 | -3.5 + 2.5 = -1
3 | -1 + 2.5 = 1.5
4 | 1.5 + 2.5 = 4
5 | 4 + 2.5 = 6.5
...
To graph the function, we plot the values from the table of values on a coordinate plane.
The explicit function can be found by noticing the pattern in the recursive formula. Each term is obtained by adding 2.5 to the previous term. This suggests a linear relationship.
Let's assume the explicit function is of the form:
a_n = m*n + b
To find the values of m and b, we can substitute a_0 = -6 into the explicit function equation:
-6 = m*0 + b
b = -6
Therefore, the explicit function for the recursive formula a_n = a_(n-1) + 2.5 is:
a_n = 2.5n - 6
To graph the recursive function and complete the table of values, we can follow these steps:
Step 1: Start with the initial value. In this case, 푎₀ = -6.
Step 2: Use the recursive formula 푎𝑛 = 푎𝑛₋₁ + 2.5 to find the next value. Repeat this step to find the subsequent values.
Step 3: Create a table to record the values of 푛 and 푎𝑛.
Step 4: Plot the values on a graph, where 푛 represents the x-axis, and 푎𝑛 represents the y-axis.
Let's start by finding values for 푎𝑛 and completing the table:
푛 | 푎𝑛
---------
0 | -6
1 | -6 + 2.5 = -3.5
2 | -3.5 + 2.5 = -1
3 | -1 + 2.5 = 1.5
4 | 1.5 + 2.5 = 4
5 | 4 + 2.5 = 6.5
...
Now, we can plot these values on a graph:
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+ + + + + + + + + + + + + + + + + + + + + + + +
| +
+ + + + + + + + +
+ +
+ +
+ + + +
Here's how you can rewrite the recursive function as an explicit function:
The given recursive function can be represented as 푎𝑛 = 푎₀ + 2.5𝑛. This equation shows that 푎𝑛 is the initial value 푎₀, added to 2.5 multiplied by 𝑛. This explicit function represents the relationship between 푎𝑛 and 푛 directly, without relying on the previous terms in the sequence.
Now, you can use this equation to find any term in the sequence directly, without having to calculate all the previous terms.
To graph the recursive function 푎₀ = -6 and 푎ₙ = 푎ₙ₋₁ + 2.5, we need to find the values of 푎 for different values of 푛.
First, let's complete the table of values:
푛 | 푎ₙ
---------
0 | -6
1 | -6 + 2.5 = -3.5
2 | -3.5 + 2.5 = -1
3 | -1 + 2.5 = 1.5
4 | 1.5 + 2.5 = 4
Now, let's plot these values on a graph. We'll represent 푛 on the x-axis and 푎ₙ on the y-axis.
(0, -6), (1, -3.5), (2, -1), (3, 1.5), (4, 4)
The graph will have these points:
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5 |
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4 | X
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3 |
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2 |
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1 |
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_____________________
0 1 2 3 4
To rewrite the recursive function as an explicit function, we need to find a formula that allows us to directly calculate 푎ₙ using 푛.
Looking at the table of values, we can observe that 푎ₙ increases by 2.5 for each 푛 value.
Therefore, the explicit function can be written as 푎ₙ = -6 + 2.5푛, which is the slope-intercept form.