State the x and y intercepts for each function below. Show all of your work.
1
y= ---- + 3
x-2
To find the x-intercept, we set y equal to zero and solve for x. To find the y-intercept, we set x equal to zero and solve for y.
For the given function:
y = 1/(x-2) + 3
To find the x-intercept, we set y = 0 and solve for x:
0 = 1/(x-2) + 3
Subtracting 3 from both sides, we get:
-3 = 1/(x-2)
Now, we can take the reciprocal of both sides:
-1/3 = x - 2
Adding 2 to both sides:
2 - 1/3 = x
Multiplying 2 by 3/3 to make it have the same denominator as 1/3:
6/3 - 1/3 = x
5/3 = x
So the x-intercept is (5/3, 0).
To find the y-intercept, we set x = 0 and solve for y:
y = 1/(0 - 2) + 3
Simplifying:
y = 1/(-2) + 3
y = -1/2 + 3
Bringing -1/2 to a common denominator of 2:
y = -1/2 + 6/2
y = 5/2
Thus, the y-intercept is (0, 5/2).
Therefore, the x-intercept is (5/3, 0) and the y-intercept is (0, 5/2).