Question

Need help finding the end behavior !

Answer 1

Let's focus on f(x) = |x| for now.

Recall that the absolute value of any number is never negative.

Some examples: |-7| = 7 and |5| = 5

So as x gets bigger in the positive direction, so does y. That explains the notation $x \to \infty, \ f(x) \to \infty$. Informally, we can say "the graph rises to the right".

Similarly, we have $x \to -\infty, \ f(x) \to \infty$ which means it "rises to the left". Both endpoints rise to positive infinity. The left side of the graph goes up forever because again the result of any absolute value function is never negative. So if we plug in say negative a million, then the result is positive a million. In a sense, the V shape absolute value function is almost like a parabola. Both have the exact same end behavior on both sides.

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Now let's move onto g(x) = 2x^2+4

The only thing that matters when determining the end behavior is the leading term. The leading term here is 2x^2

The even exponent means the endpoints either A) go up together or B) go down together. We go with case A because the leading coefficient is positive. Like I mentioned earlier, this parabola mimics the V shaped absolute value graph in terms of the end behaviors being the same.

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Lastly, let's focus on h(y) = 3y^4-2

I'm not sure why your teacher is using y when the others were using x. I'll just swap y for x to get h(x) = 3x^4-2

Like the g(x) function, the largest exponent is even, so the left and right end behaviors go in the same direction. The positive leading coefficient means we have the endpoints going upward toward positive infinity.