Question

Describe the end behavior for the graph of y=-x³- ax + 3, if a is a constant.

Answer 1

Answer: Choice C

As
`\text{x} \to \infty` then
`\text{y} \to -\infty`

As
`\text{x} \to -\infty` then
`\text{y} \to \infty`

Check out the diagram below.

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Step-by-step explanation:

The leading term is the term with the largest exponent. That would be the term of
`-\text{x}^3`. The leading term directly determines the end behavior. The other terms will not play a role.

What this means is that the end behavior of
`\text{y} = -\text{x}^3 - \text{ax} + 3` is the exact same as the end behavior of
`\text{y} = -\text{x}^3`

As x gets really big in the positive direction, we'll have
`-\text{x}^3` get really small in the negative direction. For instance, x = 10 leads to
`-\text{x}^3 = -10^3 = -1000` and x = 100 leads to
`-\text{x}^3 = -100^3 = -1,000,000`

Therefore, as
`\text{x} \to \infty` then
`\text{y} \to -\infty`

Visually the graph goes forever downward when we move to the right. We can consider this as "falls to the right".

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The graph "rises to the left" because as
`\text{x} \to -\infty` then
`\text{y} \to \infty`

Here's a small table to give numeric examples:

`\begin{array}c \cline{1-2}x & y\\\cline{1-2}-1 & 1\\\cline{1-2}-10 & 1000\\\cline{1-2}-100 & 1,000,000\\\cline{1-2}\end{array}`

As x gets more negative, y becomes more positive. We have this opposite nature going on similar to the previous section.

This graph goes uphill when moving to the left.

Check out the graph below. In the diagram I used a = 2, but you could use any value of 'a' you want to get the same end behavior.