find the slant asymptote of the graph of the rational function and use the slant asymptote to graph.
f(x)=x^2+x-2/x-7
find slant asymptote of graph of f
use slant asymptote to graph rational function
find x and y intercepts
find vertical and horizontal asymptote
To find the slant asymptote of the rational function f(x) = (x^2 + x - 2)/(x - 7), you can use long division or synthetic division.
1. Long division:
Perform long division by dividing the numerator (x^2 + x - 2) by the denominator (x - 7) to find the quotient and remainder.
x + 8
______________
x - 7 | x^2 + x - 2
- (x^2 - 7x)
8x - 2
- (8x - 56)
54
The quotient is x + 8 and the remainder is 54. This means that the slant asymptote is the line y = x + 8.
2. Synthetic division:
If you prefer using synthetic division, set up the divisions as follows:
7 | 1 1 -2
- 7 -42
__________
1 -6 -44
The last term in the bottom row (-44) represents the remainder, and the other coefficients in the bottom row (-6 and 1) represent the coefficients of the slant asymptote, which is y = -6x + 1. However, this is equivalent to y = x + 8.
To graph the rational function using the slant asymptote:
1. Plot the slant asymptote, which is the line y = x + 8. This line will be a guideline for the graph.
2. Find the x-intercept(s) by setting f(x) = 0 and solving for x. In this case, set (x^2 + x - 2)/(x - 7) = 0 and solve for x. The x-intercept(s) will be the solution(s) to this equation.
(x^2 + x - 2)/(x - 7) = 0
x^2 + x - 2 = 0
Solve this quadratic equation using factoring, completing the square, or the quadratic formula to find the x-intercept(s).
3. Find the y-intercept by substituting x = 0 into the function f(x). In this case, f(0) = (0^2 + 0 - 2)/(0 - 7) = -2/(-7) = 2/7. Therefore, the y-intercept is (0, 2/7).
4. Plot the x-intercepts and the y-intercept on your graph.
5. Find the vertical asymptote(s) by setting the denominator equal to zero and solving for x. In this case, set x - 7 = 0 and solve for x. The vertical asymptote(s) will be the solution(s) to this equation. The vertical asymptote of f(x) = (x^2 + x - 2)/(x - 7) is x = 7.
6. Determine the behavior of the function as x approaches infinity and negative infinity. As x approaches infinity or negative infinity, the function approaches the slant asymptote y = x + 8. You can plot a few additional points on either side of the graph to illustrate this behavior.
By following these steps, you should be able to graph the rational function f(x) = (x^2 + x - 2)/(x - 7) using the slant asymptote y = x + 8.