Questions: PreCalc - Lesson Quiz 6.3 Given the rational function y=(2 x^2+15+3)/(r^4+1), this either has a horizontal asymptote or a slant asymptote. Determine which one. Then enter the correct value in the blank. For the one that doesn't exist enter 'na' in the blanks. Horizontal: y= Slant: y= 7.

Solution Steps

The given rational function is: \[ y = \frac{2x^{2} + 15x + 3}{x^{4} + 1} \]

• The degree of the numerator \( 2x^{2} + 15x + 3 \) is 2.
• The degree of the denominator \( x^{4} + 1 \) is 4.

For a rational function \( y = \frac{P(x)}{Q(x)} \):

• If the degree of \( P(x) \) is less than the degree of \( Q(x) \), the horizontal asymptote is \( y = 0 \).
• If the degree of \( P(x) \) is equal to the degree of \( Q(x) \), the horizontal asymptote is \( y = \frac{\text{leading coefficient of } P(x)}{\text{leading coefficient of } Q(x)} \).
• If the degree of \( P(x) \) is greater than the degree of \( Q(x) \) by exactly 1, there is a slant asymptote.

In this case, the degree of the numerator (2) is less than the degree of the denominator (4). Therefore, there is a horizontal asymptote at \( y = 0 \), and there is no slant asymptote.

• Horizontal asymptote: \( y = 0 \).
• Slant asymptote: Does not exist, so we write "na".

Final Answer