Sketch the graph in the spaces below.

1. Identify the shape, asymptotes and quadrants for the sketch.

For $y=\frac{4}{x}$:

• $x\neq 0$, so the graph has vertical asymptote $x=0$.
• As $|x|$ becomes large, $y$ approaches $0$, so the graph has horizontal asymptote $y=0$.
• When $x>0$, $y>0$, so one branch is in Quadrant I.
• When $x<0$, $y<0$, so the other branch is in Quadrant III.

Sketch a smooth rectangular hyperbola approaching both axes but not touching them.

1. Identify and sketch the key features of the cubic graph.

For $y=-2x^3$:

• when $x=0$, $y=0$, so the graph passes through the origin;
• when $x>0$, $y<0$, so the right branch lies below the $x$-axis;
• when $x<0$, $y>0$, so the left branch lies above the $x$-axis.

Sketch a smooth decreasing cubic curve from upper left to lower right through $(0,0)$.

1. Relate the equation to the two sketched graphs and count intersections.

$\frac{4}{x}+2x^3=0 \quad \Rightarrow \quad \frac{4}{x}=-2x^3$
So the number of solutions is the number of intersections of the graphs $y=\frac{4}{x}$ and $y=-2x^3$.

From the sketches, $y=\frac{4}{x}$ lies in Quadrants I and III, while $y=-2x^3$ lies in Quadrants II and IV, so they do not intersect.

Hence, there are $0$ solutions.