Explain how you can tell (without graphing it) that the rational function
r(x)= x^6 +10 / x^4+8x^2+15
has no x intercept and no horizontal, vertical, or slant asymptote. What is its end behaviour?
Please help? thankq
Horizontal/slant intercepts:
divide the leading coefficient of the numerator by that of the denominator:
q=x^6/x^4=x²
If q is a numerical constant, the horizontal asymptote is at y=q.
If q is a linear term, such as 2x, then there is a slant asymptote along the line y=2x.
Vertical asymptotes occur where the denominator becomes zero.
Substitute y=x² in the denominator and solve for the resulting quadratic where y=-3 or -5. Clearly the solutions for x in y=-3 or -5 are complex, therefore the denominator does not become zero, hence no vertical asymptote.
To determine whether a rational function has x-intercepts and asymptotes, you can analyze its numerator and denominator polynomials separately.
First, let's analyze the numerator:
The numerator of the rational function is x^6 + 10. In order for the rational function to have an x-intercept, the numerator would need to be equal to zero. However, the polynomial x^6 + 10 has no real roots (x-values that make it equal to zero). This means that the rational function r(x) does not have any x-intercepts.
Next, let's analyze the denominator:
The denominator of the rational function is x^4 + 8x^2 + 15. To find any possible asymptotes, we need to analyze its factors. However, the quadratic x^4 + 8x^2 + 15 does not factorize into linear terms or have any real roots. Therefore, there are no vertical asymptotes.
For horizontal or slant asymptotes, we need to compare the degrees of the numerator and denominator. In this case, the highest degree in the numerator is 6, and the highest degree in the denominator is 4. Since the degree of the numerator is greater than the degree of the denominator, there are no horizontal or slant asymptotes.
In terms of end behavior, we can look at the highest degree terms in the numerator and denominator. Since the numerator has a degree of 6 and the denominator has a degree of 4, as x goes to positive or negative infinity, the denominator grows faster than the numerator. This means that the function approaches zero as x approaches infinity in both directions.
In summary:
- The rational function r(x) = (x^6 + 10) / (x^4 + 8x^2 + 15) has no x-intercepts.
- There are no vertical asymptotes.
- There are no horizontal or slant asymptotes.
- The end behavior of the function indicates that it approaches zero as x approaches positive or negative infinity.