Question

graph each proportional relationship. a. y = 5/x a. choose the correct graph below.

Explanation:

1. Recall the form of a proportional - relationship:

• A proportional relationship is of the form \(y = kx\), where \(k\) is a non - zero constant. The graph of a proportional relationship is a straight line that passes through the origin \((0,0)\).

1. Analyze the given equation \(y=\frac{5}{x}\):

• This is an inverse - variation equation, not a proportional relationship. The general form of an inverse - variation is \(y=\frac{k}{x}\), where \(k = 5\) in this case.
• For an inverse - variation \(y=\frac{k}{x}\), when \(x = 1\), \(y = 5\); when \(x = 5\), \(y = 1\).
• The graph of \(y=\frac{5}{x}\) is a hyperbola. It has two branches, one in the first quadrant and one in the third quadrant for \(k>0\).
• Since the question asks to graph a proportional relationship and \(y=\frac{5}{x}\) is not a proportional relationship, there seems to be a mis - match in the problem statement. But if we assume we are just graphing \(y=\frac{5}{x}\):
• We can find some points:
• When \(x = 1\), \(y = 5\); when \(x = 5\), \(y = 1\); when \(x=-1\), \(y=-5\); when \(x = - 5\), \(y=-1\).
• However, if we assume the problem was meant to be about a proportional relationship (despite the given equation being inverse - variation), a proportional relationship \(y = kx\) has the following properties for graphing:
• The line passes through the origin \((0,0)\). If \(k>0\), the line has a positive slope. If \(k<0\), the line has a negative slope.
• Since there is no correct graph for \(y=\frac{5}{x}\) among the options for a proportional relationship graph (as it is not a proportional relationship), and no correct equation for the given graphs (assuming they are for proportional relationships), we need to re - check the problem. But if we ignore the mis - match and just analyze the form of \(y=\frac{5}{x}\):
• We know that as \(x\) approaches \(0\) from the positive side, \(y\) approaches \(+\infty\), and as \(x\) approaches \(0\) from the negative side, \(y\) approaches \(-\infty\).
• None of the given graphs (which seem to be for linear relationships passing through the origin for proportional relationships) are correct for \(y=\frac{5}{x}\). But if we assume we made a wrong start and the equation was supposed to be \(y = 5x\) (a proportional relationship):
• The slope \(k = 5>0\), and the line passes through the origin \((0,0)\). We can find points: when \(x = 1\), \(y = 5\); when \(x=-1\), \(y=-5\).
• A proportional relationship \(y = kx\) has a straight - line graph passing through the origin.

Since the problem seems to have a mix - up, if we assume we are graphing a proportional relationship \(y = 5x\):

Step 1: Identify the form of a proportional relationship

A proportional relationship is \(y = kx\), and for \(y = 5x\), \(k = 5>0\), so the line has a positive slope and passes through \((0,0)\).

Step 2: Find some points

When \(x = 1\), \(y=5\times1 = 5\); when \(x=-1\), \(y=5\times(-1)=-5\).

Answer:

There is no correct answer among the given options as the equation \(y=\frac{5}{x}\) is not a proportional relationship. If the equation was meant to be \(y = 5x\), a graph with a straight line passing through the origin and having a positive slope would be correct, but no such graph is shown clearly among the provided options.