Question

which graph represents the function $f(x) = \frac{4}{x}$? four graphs are shown, each with a coordinate grid and a red curve

Explanation:

Step1: Analyze the function type

The function \( f(x)=\frac{4}{x} \) is a rational function, specifically a hyperbola. For \( y = \frac{k}{x} \) (where \( k = 4>0 \)), the graph has two branches: one in the first quadrant (where \( x>0,y>0 \)) and one in the third quadrant (where \( x<0,y<0 \)) because when \( x \) and \( y \) have the same sign, their product is positive (equal to \( k \)).

Step2: Check the quadrants of each graph

• First graph: One branch in the first quadrant and one in the fourth/second? Wait, no, let's re - check. Wait, the first graph: when \( x>0 \), \( y \) is positive (first quadrant), when \( x<0 \), \( y \) is negative (third quadrant? Wait, no, the first graph's left - hand branch: when \( x<0 \), \( y \) is negative? Wait, no, let's look at the second graph. Wait, the function \( y=\frac{4}{x} \), when \( x = 1 \), \( y = 4 \); when \( x=-1 \), \( y=-4 \). So the two branches should be in the first ( \( x>0,y>0 \)) and third ( \( x<0,y<0 \)) quadrants.

Looking at the graphs:

• The second graph: The right - hand branch is in the first quadrant ( \( x>0,y>0 \)) and the left - hand branch is in the third quadrant ( \( x<0,y<0 \)), which matches the sign behavior of \( y = \frac{4}{x} \) (since \( k = 4>0 \), \( x \) and \( y \) have the same sign).
• The first graph: The left - hand branch seems to be in the second/third? Wait, no, when \( x<0 \), for \( y=\frac{4}{x} \), \( y \) should be negative. The first graph's left - hand branch: when \( x<0 \), \( y \) is negative? Wait, no, the first graph's left - hand branch: let's take \( x=-1 \), if the graph is \( y=\frac{4}{x} \), \( y=-4 \), but the first graph's left - hand branch at \( x=-1 \) seems to be at \( y=-2 \)? No, maybe I misread. Wait, the second graph: when \( x = 1 \), \( y = 4 \) (matches \( y=\frac{4}{x} \)), when \( x=-1 \), \( y=-4 \) (matches). The other graphs: the third and fourth graphs have branches in second - fourth quadrants (where \( x \) and \( y \) have opposite signs, which would be for \( k<0 \)). The first graph: let's check the sign. For \( y=\frac{4}{x} \), \( x \) and \( y \) must have the same sign. The second graph has both branches in quadrants where \( x \) and \( y \) have the same sign (first and third), which is correct for \( k = 4>0 \).

Answer:

The Middle Graph (the second graph among the three top - row graphs, or the one with branches in the first and third quadrants)