For the rational function below (i) Find the intercepts for the graph; (ii) Determine the domain; (iii) Find any vertical or
horizontal asymptotes for the graph; (iv) Sketch any asymptotes as dashed lines. Then sketch the graph of y = f(x).
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A) (ii)
B) (ii)
C) (ii)
D) (ii)

Question

For the rational function below (i) Find the intercepts for the graph; (ii) Determine the domain; (iii) Find any vertical or
 horizontal asymptotes for the graph; (iv) Sketch any asymptotes as dashed lines. Then sketch the graph of y = f(x).
-
A) (ii) 
B) (ii) 
C) (ii) 
D) (ii)

Quizplus · Accepted Answer

(i) The x-intercept can be found by setting y=0 and solving for x:
0 = (x-2)/(x+4)
x - 2 = 0
x = 2
Therefore, the x-intercept is (2,0). The graph does not have a y-intercept.

(ii) The domain is all real numbers except x=-4, since this value would make the denominator of the fraction zero and the function undefined.

(iii) The vertical asymptote can be found by setting the denominator of the fraction equal to zero:
x + 4 = 0
x = -4
Therefore, there is a vertical asymptote at x=-4. There is no horizontal asymptote.

(iv) Sketching the vertical asymptote as a dashed line at x=-4 and the x-axis intercept at (2,0), we can then connect these two points with a curve that approaches the vertical asymptote but never actually touches it. The resulting graph should resemble a hyperbola. (i) Intercepts: 
- y-intercept: To find the y-intercept, we let x = 0. Plugging in x = 0 into the equation, we get y = -3/4. So the y-intercept is (0, -3/4).
- x-intercept: To find the x-intercept, we let y = 0. Plugging in y = 0 into the equation, we get -x^2 + 4x = 0. Factoring out x, we get x(-x + 4) = 0. So the x-intercepts are (0, 0) and (4, 0).

(ii) Domain: 
The domain is all real numbers except x = 0 and x = 4, since the function is undefined at those points.

(iii) Asymptotes: 
- Vertical asymptote: There is a vertical asymptote at x = 0 since the denominator becomes 0 and the function goes to infinity as x approaches 0 from both sides.
- Horizontal asymptote: There is a horizontal asymptote at y = 0 since as x becomes very large (positive or negative), the leading term of the function (-x^2) dominates and the function approaches 0.

(iv) Sketch: 
- Vertical asymptote: Draw a dashed line at x = 0.
- Horizontal asymptote: Draw a dashed line at y = 0.
- Intercepts: Mark the y-intercept at (0, -3/4) and the x-intercepts at (0, 0) and (4, 0).
- Shape of the graph: Since the leading term of the function is -x^2, the graph will be a downward-facing parabola. As x goes to infinity, the parabola gets narrower and closer to the x-axis, but never touches it. Similarly, as x goes to negative infinity, the parabola gets narrower and closer to the x-axis, but never touches it. The graph passes through the y-axis at (0, -3/4), but does not touch the x-axis.
Overall, the graph looks like a sideways "U" shape, with a vertical asymptote at x = 0 and a horizontal asymptote at y = 0.

For the Rational Function Below (I) Find the Intercepts for the Graph;