Question $f(x)=7 x^{2}(x+3)(x+6)$
Describe the behavior of the graph of $f(x)$ as $x \rightarrow \pm \infty$ .

Studdy Solution

STEP 1

What is this asking? Where does the graph go when $x$ gets really big or really small? Watch out! Don't forget to consider both positive and negative infinity for $x$ , and how the signs interact!

STEP 2

1. Expand the function
2. Analyze end behavior

STEP 3

Alright, let's expand this function!
We'll multiply the terms together step-by-step.
First, we'll multiply $(x+3)$ and $(x+6)$ : $(x+3)(x+6) = x(x+6) + 3(x+6) = x^2 + 6x + 3x + 18 = x^2 + 9x + 18$ So, $f(x) = 7x^2(x^2 + 9x + 18)$ .

STEP 4

Now, let's distribute the $7x^2$ : $7x^2(x^2 + 9x + 18) = 7x^2 \cdot x^2 + 7x^2 \cdot 9x + 7x^2 \cdot 18 = 7x^4 + 63x^3 + 126x^2$ Awesome! Our expanded function is $f(x) = 7x^4 + 63x^3 + 126x^2$ .

STEP 5

As $x$ gets super duper large (approaching positive infinity), the term with the highest power, $7x^4$ , dominates the function's behavior.
Since $x^4$ will be a massive positive number, and we're multiplying it by positive 7, $f(x)$ will also approach positive infinity.
Think of it like this: the other terms, $63x^3$ and $126x^2$ , are still getting large, but $7x^4$ is growing so much faster that it takes over!

STEP 6

Now, what happens when $x$ approaches negative infinity?
Well, $x^4$ will still be a massive positive number because any negative number raised to an even power becomes positive.
So, $7x^4$ will again be a huge positive number.
The other terms, even though they have odd powers, won't grow fast enough to change the overall behavior.
Therefore, as $x$ approaches negative infinity, $f(x)$ also approaches positive infinity!

STEP 7

As $x \rightarrow \infty$ , $f(x) \rightarrow \infty$ .
As $x \rightarrow -\infty$ , $f(x) \rightarrow \infty$ .

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