Find the properties of the following rational function.
f(x)=(x+7)(x-2)/x^2-6x+8
x intercept:
y intercept:
Hole:
H.A: also answer if this is x or y
V.A: also answer if this is x or y
To find the properties of the rational function f(x), we need to first determine its x-intercept, y-intercept, hole, horizontal asymptote, and vertical asymptote.
1. X-Intercept:
To find the x-intercept, we set y = 0 and solve for x.
0 = (x+7)(x-2)/(x^2-6x+8)
(x+7)(x-2) = 0
This gives x = -7 or x = 2 as the x-intercepts.
2. Y-Intercept:
To find the y-intercept, we set x = 0 and solve for y.
y = (0+7)(0-2)/(0^2-6*0+8)
y = 14/8
y = 7/4
Thus, the y-intercept is (0, 7/4).
3. Hole:
To find the hole, we need to simplify the rational function.
f(x) = (x+7)(x-2)/(x^2-6x+8)
f(x) = (x+7)(x-2)/((x-4)(x-2))
Since there is a common factor of (x-2) in both the numerator and denominator, we have a hole at x = 2.
4. Horizontal Asymptote:
The horizontal asymptote is found by looking at the leading terms of the numerator and denominator as x approaches positive or negative infinity. In this case, the degree of the numerator is equal to the degree of the denominator, so the horizontal asymptote is y = a/b, where a is the coefficient of the leading term in the numerator and b is the coefficient of the leading term in the denominator. Therefore, the horizontal asymptote is y = 1.
5. Vertical Asymptote:
To find the vertical asymptote, we need to find the values of x that make the denominator equal to zero. The denominator is x^2-6x+8 and factored form of this is (x-4)(x-2). Therefore, the vertical asymptote is x = 4.
In summary:
- X-Intercepts: x = -7, x = 2
- Y-Intercept: (0, 7/4)
- Hole: x = 2
- Horizontal Asymptote: y = 1
- Vertical Asymptote: x = 4
The x intercept, the y intercept and the hole are wrong try again
I apologize for the mistake. Let's correct those calculations:
1. X-Intercept:
To find the x-intercept, we set y = 0 and solve for x.
0 = (x+7)(x-2)/(x^2-6x+8)
(x+7)(x-2) = 0
This gives x = -7 or x = 2 as the x-intercepts.
2. Y-Intercept:
To find the y-intercept, we set x = 0 and solve for y.
y = (0+7)(0-2)/(0^2-6*0+8)
y = -14/8
y = -7/4
Thus, the y-intercept is (0, -7/4).
3. Hole:
To find the hole, we need to simplify the rational function.
f(x) = (x+7)(x-2)/(x^2-6x+8)
f(x) = (x+7)(x-2)/((x-4)(x-2))
Since there is a common factor of (x-2) in both the numerator and denominator, we have a hole at x = 2.
Therefore, the correct x-intercepts are x = -7 and x = 2, the y-intercept is (0, -7/4), and there is a hole at x = 2. Thank you for pointing out the mistake.