Find the horizontal and vertical asymptotes of the curve.
y = 3x+6/x−7
To find the horizontal and vertical asymptotes of the curve, you need to analyze the behavior of the function as x approaches positive or negative infinity.
First, let's find the horizontal asymptote. The horizontal asymptote indicates the value that y approaches as x approaches infinity or negative infinity.
To find the horizontal asymptote, we need to compare the degrees of the numerator and denominator of the function. In this case, the numerator is a linear function (degree 1) and the denominator is also a linear function (degree 1).
When the degrees of the numerator and denominator are the same, you can find the horizontal asymptote by taking the ratio of the leading coefficients. In this case, the leading coefficients are 3 and 1.
Therefore, the horizontal asymptote for the given function is y = 3/1 or simply y = 3.
Next, let's find the vertical asymptote. The vertical asymptote indicates the value that x approaches as it approaches a certain value.
To find the vertical asymptote, we need to determine where the function is undefined. In this case, the function is undefined when the denominator equals zero, as division by zero is not allowed.
Setting the denominator equal to zero, we have:
x - 7 = 0
Solving for x, we find that x = 7.
Therefore, the vertical asymptote for the given function is x = 7.