analyze the graph of the following function.
r(x)= x^2+x-6
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x^2-x-2
r(x) = Y = x^2 + x - 6
C = -6 = -1(6) = -2(3).
Select the pair of factors whose sum = 1(B):
Y = (x-2)(x+3) = 0
Solution set: X = -2, and 3 = x-Intercepts.
Axis: x = Xv = -B/2A = -1/2. = h.
k = Yv = (1/2)^2 -1/2 - 6
k = -6.25.
V(h,k) = V(-0.5,-6.25)
The following points can be used for graphing:
(x,y).
(-3,0)
(-2,-4)
(-1,-6)
V(-0.5,-6.25)
(1,-4)
(2,0)
(3,6)
Correction:
Solution set: X = -3, and 2.
To analyze the graph of the function r(x) = (x^2 + x - 6)/(x^2 - x - 2), you can start by finding the domain and vertical asymptotes, determining the x-intercepts and y-intercepts, finding the behavior towards positive and negative infinity, and checking for any horizontal asymptotes.
1. Domain and Vertical Asymptotes:
To find the domain, we need to consider any values of x that would make the denominator of the function equal to zero. In this case, the denominator is x^2 - x - 2 = 0. We can factor this equation as (x - 2)(x + 1) = 0. Therefore, x cannot be -1 or 2.
2. X-intercepts:
To find the x-intercepts, we need to set the numerator equal to zero: x^2 + x - 6 = 0. We can factor this equation as (x - 2)(x + 3) = 0. Hence, the x-intercepts are x = 2 and x = -3.
3. Y-intercept:
To find the y-intercept, we substitute x = 0 into the function: r(0) = (0^2 + 0 - 6)/(0^2 - 0 - 2) = -6/-2 = 3. Therefore, the y-intercept is 3.
4. Behavior towards Positive and Negative Infinity:
As x approaches positive infinity (x → +∞), the expression in the numerator and denominator becomes dominated by the highest power of x (x^2). So, r(x) becomes x^2/x^2 = 1 as x gets large.
Similarly, as x approaches negative infinity (x → -∞), the expression in the numerator and denominator again becomes dominated by x^2. So, r(x) becomes x^2/x^2 = 1 as x becomes highly negative.
5. Horizontal Asymptotes:
To check for horizontal asymptotes, we find the limit of the function as x approaches either positive or negative infinity. In this case, as mentioned earlier, both limits are equal to 1. Therefore, there are no horizontal asymptotes.
By considering all the above information, you can now plot the graph of r(x) = (x^2 + x - 6)/(x^2 - x - 2) using the x-intercepts, y-intercept, and the behavior towards infinity.