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Which is the graph of y equals StartFraction k over x EndFraction, where k is a constant?

Answer options with 5 options
A.
On a coordinate plane, a parabola opens up. It goes through (negative 4, 8), has a vertex at (0, 0), and goes through (4, 8).

D.
A graph.Short description, A graph.,Long description,
On a coordinate plane, a curve opens up and left in quadrants 1 and 2. The curve is asymptotic to the negative x-axis, and curves up sharply, going through (2, 8).

B.
A graph.Short description, A graph.,Long description,
On a coordinate plane, a curve opens left in quadrant 2 and another curve opens right in quadrant 1. Both curves are asymptotic to the positive y-axis and have a horizontal asymptote at y equals 2.

E.
A graph.Short description, A graph.,Long description,
On a coordinate plane, a hyperbola with center at the origin has a curve in quadrant 2 and a curve in quadrant 4. One curve is asymptotic to the negative x-axis and the positive y-axis. The other curve is asymptotic to the positive x-axis and the negative y-axis

C.
A graph.Short description, A graph.,Long description,
On a coordinate plane, a curve opens right in quadrants 1 and 4. It starts at the top of the graph, crosses the x-axis at (4, 0), and goes through (8, negative 4).

1 answer

The equation \( y = \frac{k}{x} \) represents a hyperbola, where the shape of the graph depends on the sign of the constant \( k \). If \( k > 0 \), the graph has two branches, one in the first quadrant and the other in the third quadrant. If \( k < 0 \), the branches are located in the second and fourth quadrants.

Given these characteristics, we can analyze the provided options:

  • Option A describes a parabola, which does not fit the equation \( y = \frac{k}{x} \).
  • Option D also does not represent \( y = \frac{k}{x} \), as it suggests a curve that opens up and left.
  • Option B mentions curves in quadrants 1 and 2 with horizontal asymptotes, which does not match the format of a hyperbola defined by \( y = \frac{k}{x} \).
  • Option C describes a curve that opens right in quadrants 1 and 4, which also does not match the hyperbolic nature of \( y = \frac{k}{x} \).
  • Option E correctly describes a hyperbola with one curve in quadrant 2 and another in quadrant 4, asymptotic to the x-axis and y-axis, which fits the behavior of the graph of \( y = \frac{k}{x} \) where \( k \) can be either positive or negative.

Therefore, the correct answer is:

E. On a coordinate plane, a hyperbola with center at the origin has a curve in quadrant 2 and a curve in quadrant 4. One curve is asymptotic to the negative x-axis and the positive y-axis. The other curve is asymptotic to the positive x-axis and the negative y-axis.