analyze the graph of the function
r(x)= 13x + 13
________
5x + 15
what can you figure out? Consider zeros, asymptotes of all kinds, maxima and minima.
So, there is not one in particular that this question falls under then. That's where I have my problem is not knowing what and where to start from as far as the formulas for each equation goes. I did good with algebra but come to this my brain has just stopped. Like I said ealier I know I do not need this class for nursing but still.
since the numerator is 13x+13, r(x) = 0 when x = -1
since the denominator is 5x+15, the denominator is zero when x = -3, so there is a vertical asymptote at x = -3.
As x gets huge, r(x) =~ 13x/5x = 13/5, so there is a horizontal asymptote at y = 13/5
To analyze the graph of the function r(x) = (13x + 13)/(5x + 15), we can start by identifying the key components: the numerator and denominator of the function.
1. Vertical Asymptotes:
The vertical asymptotes occur when the denominator of the function equals zero. To find these points, we set 5x + 15 = 0 and solve for x:
5x + 15 = 0
5x = -15
x = -3
So, there is a vertical asymptote at x = -3.
2. Horizontal Asymptotes:
To determine the horizontal asymptote(s), we compare the degrees of the numerator and denominator:
- The degree of the numerator is 1 (x^1).
- The degree of the denominator is also 1 (x^1).
Since the degrees are the same, we divide the leading coefficients (the coefficients in front of the highest power of x):
Leading coefficient of numerator: 13
Leading coefficient of denominator: 5
The ratio of the leading coefficients is 13/5. Therefore, the horizontal asymptote is y = 13/5.
3. x-Intercept:
To find the x-intercept(s), we set r(x) = 0 and solve for x:
(13x + 13)/(5x + 15) = 0
13x + 13 = 0
Solving for x, we obtain:
13x = -13
x = -1
So, the x-intercept is at x = -1.
4. y-Intercept:
To find the y-intercept, we substitute x = 0 into the function:
r(0) = (13(0) + 13)/(5(0) + 15)
r(0) = 13/15
Thus, the y-intercept is at y = 13/15.
By analyzing these key components, we can now plot the graph of the function r(x)=(13x + 13)/(5x + 15) and observe its behavior around the asymptotes and intercepts.