find the slant asymptote of the graphof the rational function.
1.f(x)=x^2-x-2/x-7
y=?
2. determine the symmetry of the graph of f.
a. the graph has y-axis symmetry f(-x)=f(x)
b. the graph has orgin symmetry f(-x)= -f(x)
c. the graph has both y-axs and orgin symmetry
d. the graph has neither y-axis nor orgin symmetry
3. find the y-intercepts
4. find the x-intercepts
5. find the vertical intercepts
6. find the horizontal asymptotes
please show work
since f(x) = x + 6 + 40/(x-7)
f(x) = x+6 plus an amount that gets small as x gets large.
So, the slant asymptote is y=x+6
go to wolframalpha.com and type in
(x^2-x-2)/(x-7)
and it will show the graph, making the other answers easy to decide.
To find the slant asymptote of the rational function f(x) = (x^2-x-2)/(x-7), you can use polynomial long division:
Step 1: Divide the numerator (x^2 - x - 2) by the denominator (x - 7).
x - 8
___________
x - 7 | x^2 - x - 2
- (x^2 - 7x)
___________
6x - 2
- (6x - 42)
___________
40
Step 2: Rewrite the original function as the sum of the quotient and a remainder over the denominator:
f(x) = (x - 8) + (40/(x - 7))
The quotient is x - 8, and the remainder is 40.
Therefore, the slant asymptote of the graph is y = x - 8.
Now, let's address the other questions:
2. To determine the symmetry of the graph of f, we need to check if f(-x) = f(x) or f(-x) = -f(x).
a. If f(-x) = f(x) for all x, then the graph has y-axis symmetry.
b. If f(-x) = -f(x) for all x, then the graph has origin symmetry.
c. If both conditions a and b above are satisfied, then the graph has both y-axis and origin symmetry.
d. If neither condition a nor b is satisfied, then the graph has neither y-axis nor origin symmetry.
To check the symmetry for the given function, we substitute -x for x in the expression f(x) = (x^2-x-2)/(x-7) and simplify.
a. If f(-x) = f(x), then the function has y-axis symmetry.
f(-x) = ((-x)^2 - (-x) - 2)/(-x - 7)
= (x^2 + x - 2)/(-x - 7)
Therefore, f(x) and f(-x) are not equal, so the graph does not have y-axis symmetry.
b. If f(-x) = -f(x), then the function has origin symmetry.
f(-x) = (x^2 - x - 2)/(-x - 7)
Therefore, f(x) and -f(x) are not equal, so the graph does not have origin symmetry.
d. Since none of the conditions are satisfied, the graph has neither y-axis nor origin symmetry.
3. The y-intercept is the point where the graph intersects the y-axis, which occurs when x = 0.
Substituting x = 0 into the function f(x):
f(0) = (0^2 - 0 - 2)/(0 - 7)
= (-2)/(-7) = 2/7
Therefore, the y-intercept is the point (0, 2/7).
4. The x-intercepts are the points where the graph intersects the x-axis, which occurs when f(x) = 0.
Setting the numerator x^2 - x - 2 equal to zero and solving for x:
x^2 - x - 2 = 0
(x - 2)(x + 1) = 0
x - 2 = 0 or x + 1 = 0
x = 2 or x = -1
Therefore, the x-intercepts are the points (2, 0) and (-1, 0).
5. Vertical intercepts occur when x approaches infinity or negative infinity, or when the denominator becomes zero.
The denominator, x - 7, becomes zero when x = 7.
Therefore, the vertical intercept is the point (7, f(7)).
6. To find the horizontal asymptotes, we examine the behavior of the function as x approaches infinity and negative infinity.
As x approaches infinity, the most dominant terms are x^2 in the numerator and x in the denominator.
Therefore, the degree of the numerator is greater than the degree of the denominator.
In this case, the graph has no horizontal asymptote, but instead, a slant asymptote as found previously.
As x approaches negative infinity, the most dominant terms are x^2 in the numerator and x in the denominator.
Again, the degree of the numerator is greater than the degree of the denominator, so there is no horizontal asymptote.
Therefore, the function f(x) = (x^2 - x - 2)/(x - 7) has a slant asymptote y = x - 8, no y-axis or origin symmetry, y-intercept (0, 2/7), x-intercepts (2, 0) and (-1, 0), a vertical intercept at (7, f(7)), and no horizontal asymptote.