Studdy AI Logo
Math  /  Algebra

QuestionFind the vertical asymptotes (VA) and horizontal asymptotes (HA) of the rational function y=x2+4x+32x2+5x−7 y = \frac{x^2 + 4x + 3}{2x^2 + 5x - 7} .

Studdy Solution

STEP 1

1. A vertical asymptote occurs where the denominator of a rational function equals zero, provided the numerator is not zero at those points.
2. A horizontal asymptote is determined by comparing the degrees of the numerator and the denominator.

STEP 2

1. Find the vertical asymptotes by solving for when the denominator equals zero.
2. Determine if there are horizontal asymptotes by comparing the degrees of the numerator and denominator.

STEP 3

To find the vertical asymptotes, set the denominator equal to zero and solve for x x :
2x2+5x−7=0 2x^2 + 5x - 7 = 0

STEP 4

Solve the quadratic equation 2x2+5x−7=0 2x^2 + 5x - 7 = 0 using the quadratic formula:
The quadratic formula is given by:
x=−b±b2−4ac2a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
where a=2 a = 2 , b=5 b = 5 , and c=−7 c = -7 .

STEP 5

Calculate the discriminant b2−4ac b^2 - 4ac :
b2−4ac=52−4⋅2⋅(−7) b^2 - 4ac = 5^2 - 4 \cdot 2 \cdot (-7) =25+56 = 25 + 56 =81 = 81

STEP 6

Substitute the values into the quadratic formula:
x=−5±814 x = \frac{-5 \pm \sqrt{81}}{4} x=−5±94 x = \frac{-5 \pm 9}{4}
Calculate the two possible solutions:
x=−5+94=1 x = \frac{-5 + 9}{4} = 1 x=−5−94=−144=−72 x = \frac{-5 - 9}{4} = -\frac{14}{4} = -\frac{7}{2}
The vertical asymptotes are x=1 x = 1 and x=−72 x = -\frac{7}{2} .

STEP 7

To find the horizontal asymptotes, compare the degrees of the numerator and denominator:
The degree of the numerator x2+4x+3 x^2 + 4x + 3 is 2. The degree of the denominator 2x2+5x−7 2x^2 + 5x - 7 is 2.
Since the degrees are equal, the horizontal asymptote is determined by the ratio of the leading coefficients.

STEP 8

The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is 2. Therefore, the horizontal asymptote is:
y=12 y = \frac{1}{2}
The vertical asymptotes are x=1 x = 1 and x=−72 x = -\frac{7}{2} , and the horizontal asymptote is y=12 y = \frac{1}{2} .

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord