The graph of y=1/(14x+90) has A horizontal asymptotes, B vertical asymptotes, and the graphs of the functions y = 14x + 90 and y=1/(14x+90) will intersect C times. The values of A, B, and C are...?
rational functions can have vertical asymptotes where the denominator is zero, and a horizontal asymptote.
14x+90 = 1/(14x+90)
(14x+90)^2 = 1
14x+90 = ±1
so (A,B,C) = (1,1,2)
better review the topic of rational functions. Also see the graphs at
https://www.wolframalpha.com/input/?i=14x%2B90+%3D+1%2F%2814x%2B90%29+for+-7+%3C%3D+x+%3C%3D+-5
To determine the values of A, B, and C for the graph of y = 1/(14x+90), we need to analyze the behavior of the function as x approaches certain limits.
A) Horizontal Asymptotes (A):
To find the horizontal asymptote, we observe the behavior of the function as x approaches positive or negative infinity. As x becomes very large (positive or negative), the term 14x+90 dominates the expression, making the denominator much larger than the numerator. Consequently, the fraction becomes very close to zero. Therefore, the horizontal asymptote is at y = 0. Hence, A = 0.
B) Vertical Asymptotes (B):
To find vertical asymptotes, we need to determine the x-values where the denominator of the fraction becomes zero, resulting in division by zero. In this case, solving the equation 14x+90 = 0 gives us x = -90/14 (or approximately -6.43). Hence, the vertical asymptote is at x = -6.43. Therefore, B = -6.43.
C) Intersection Points (C):
To find the intersection points, we need to equate the given function to the linear function y = 14x + 90.
Setting 1/(14x+90) = 14x + 90 leads to:
1 = (14x + 90)(14x + 90)
Expanding and simplifying the equation, we get:
0 = 196x^2 + 1260x + 8100 - 1
0 = 196x^2 + 1260x + 8099
To solve this quadratic equation, you can use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
By plugging in the values a = 196, b = 1260, and c = 8099 into the quadratic formula and evaluating, we get two possible x-values.
Using the quadratic formula, we find that x ≈ -8.262 and x ≈ -0.064.
Since it is a quadratic equation, there are two solutions, which means the graphs will intersect at two points.
Hence, C = 2.
To summarize:
A) The graph has 1 horizontal asymptote (A = 1).
B) The graph has 1 vertical asymptote (B = 1).
C) The graphs intersect at 2 points (C = 2).