Respuesta :
Answer:
As x gets smaller, pointing to negative infinity, the value of f decreses, pointing to negative infinity.
As x gets increases, pointing to positve infinity, the value of f decreses, pointing to negative infinity.
Step-by-step explanation:
To find the end behaviour of a function f(x), we calculate these following limits:
[tex]\lim_{x \to +\infty} f(x)[/tex]
And
[tex]\lim_{x \to -\infty} f(x)[/tex]
In this question:
[tex]f(x) = -4x^{6} + 6x^{2} - 52[/tex]
At negative infinity:
[tex]\lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} -4x^{6} + 6x^{2} - 52[/tex]
When the variable points to infinity, we only consider the term with the highest exponent. So
[tex]\lim_{x \to -\infty} -4x^{6} + 6x^{2} - 52 = \lim_{x \to -\infty} -4x^{6} = -4*(-\infty)^{6} = -(\infty) = -\infty[/tex]
So as x gets smaller, pointing to negative infinity, the value of f decreses, pointing to negative infinity.
Positive infinity:
[tex]\lim_{x \to \infty} f(x) = \lim_{x \to \infty} -4x^{6} + 6x^{2} - 52 = \lim_{x \to \infty} -4x^{6} = -4*(\infty)^{6} = -(\infty) = -\infty[/tex]
So as x gets increases, pointing to positve infinity, the value of f decreses, pointing to negative infinity.
f(x) = -4x^6 +6x^2-52
The leading coefficient is negative so the left end of the graph goes down.
f(x) is an even function so both ends of the graph go in the same direction.
