Let's analyze the given function and see the impact of adding each of the new terms on the graph:
### Original Function:
[tex]\[ y = -2x^7 + 5x^6 - 24 \][/tex]
#### End Behavior of the Original Function:
- The highest power term in the original function is [tex]\(-2x^7\)[/tex].
- Because 7 is an odd number and the coefficient [tex]\(-2\)[/tex] is negative, the end behavior of the original function is as follows:
- As [tex]\( x \to \infty \)[/tex], [tex]\( y \to -\infty \)[/tex]
- As [tex]\( x \to -\infty \)[/tex], [tex]\( y \to \infty \)[/tex]
### Change 1: Addition of [tex]\(-x^8\)[/tex]
New Function:
[tex]\[ y = -x^8 - 2x^7 + 5x^6 - 24 \][/tex]
#### End Behavior with Addition of [tex]\(-x^8\)[/tex]:
- The highest power term in this new function is [tex]\(-x^8\)[/tex].
- Because 8 is an even number and the coefficient [tex]\(-1\)[/tex] is negative, the end behavior of the new function changes to:
- As [tex]\( x \to \infty \)[/tex], [tex]\( y \to -\infty \)[/tex] (the function decreases indefinitely)
- As [tex]\( x \to -\infty \)[/tex], [tex]\( y \to -\infty \)[/tex] (the function decreases indefinitely)
This means the graph will decline to negative infinity in both directions along the x-axis.
### Change 2: Addition of [tex]\(5x^7\)[/tex]
New Function:
[tex]\[ y = 5x^7 - 2x^7 + 5x^6 - 24 \][/tex]
Combining like terms, we get:
[tex]\[ y = (5x^7 - 2x^7) + 5x^6 - 24 \][/tex]
[tex]\[ y = 3x^7 + 5x^6 - 24 \][/tex]
#### End Behavior with Addition of [tex]\(5x^7\)[/tex]:
- The highest power term in this new function is [tex]\(3x^7\)[/tex].
- Because 7 is an odd number and the coefficient [tex]\(3\)[/tex] is positive, the end behavior of the new function changes to:
- As [tex]\( x \to \infty \)[/tex], [tex]\( y \to \infty \)[/tex] (the function increases indefinitely)
- As [tex]\( x \to -\infty \)[/tex], [tex]\( y \to -\infty \)[/tex] (the function decreases indefinitely)
This means the graph will rise to positive infinity as [tex]\( x \)[/tex] increases and fall to negative infinity as [tex]\( x \)[/tex] decreases.
### Summary
Adding [tex]\(-x^8\)[/tex]:
- The graph declines toward [tex]\(-\infty\)[/tex] in both directions (as [tex]\( x \to \infty \)[/tex] and [tex]\( x \to -\infty \)[/tex]).
Adding [tex]\(5x^7\)[/tex]:
- The graph rises toward [tex]\(+\infty\)[/tex] as [tex]\( x \to \infty \)[/tex] and falls toward [tex]\(-\infty\)[/tex] as [tex]\( x \to -\infty \)[/tex].
These changes reflect the different dominant terms in each modified function, affecting how the graph behaves at the extremes.
