Question

tables, graphs, and equations
your teacher will assign you one of these three points:
a(5, 2), b(4, 5), c(8, 5)

1. on the coordinate plane, plot and label only your assigned point.

coordinate plane grid from (0,0) to (10,10) shown

1. graph the proportional relationship that is defined by your point. that is, use a ruler to draw a line that starts at the origin, goes through your point, and continues to the edge of the grid.
2. use your graph to find the y - value that goes with each of these x - values.

x	y
6	18

your teacher will give you a completed table. use it to check your values.

1. choose three rows, other than the row that represents the origin, from the completed table. record the values and compute \\(\frac{y}{x}\\) for each row. what do you notice about these values?

table with columns x, y, \\(\frac{y}{x}\\) shown, empty rows for recording

Explanation:

Step1: Analyze the Proportional Relationship

For a proportional relationship \( y = kx \) (where \( k \) is the constant of proportionality), the slope \( k \) can be found from the given point. Let's take point \( A(5, 2) \) first. The constant of proportionality \( k=\frac{y}{x}=\frac{2}{5} = 0.4 \). For point \( B(4, 5) \), \( k=\frac{5}{4}=1.25 \), and for point \( C(8, 5) \), \( k=\frac{5}{8} = 0.625 \). But since the problem is about a proportional relationship through the assigned point, we'll use the general method.

Step2: Find \( y \)-values for \( x = 2 \) and \( x = 6 \)

Let's assume we are assigned point \( A(5, 2) \). The equation of the line is \( y=\frac{2}{5}x \) (since \( k=\frac{2}{5} \)).

• For \( x = 2 \): \( y=\frac{2}{5}\times2=\frac{4}{5}=0.8 \)? Wait, but the handwritten answer has 6 and 18, maybe the assigned point is \( B(4, 5) \)? Wait, no, maybe I misread. Wait, the table has \( x = 2 \), \( y = 6 \) and \( x = 6 \), \( y = 18 \). Let's check the slope: \( \frac{6}{2}=3 \), \( \frac{18}{6}=3 \). So the constant of proportionality \( k = 3 \). So the equation is \( y = 3x \). Let's check the points: If \( x = 5 \), \( y = 15 \) (not \( A(5,2) \)), \( x = 4 \), \( y = 12 \) (not \( B(4,5) \)), \( x = 8 \), \( y = 24 \) (not \( C(8,5) \)). Wait, maybe the assigned point is a different one, but the key is that in a proportional relationship, \( \frac{y}{x} \) is constant.

Step3: Compute \( \frac{y}{x} \) for the table

Take the table values (assuming the table from the proportional relationship with \( k = 3 \)):

• For \( x = 2 \), \( y = 6 \): \( \frac{y}{x}=\frac{6}{2}=3 \)
• For \( x = 6 \), \( y = 18 \): \( \frac{y}{x}=\frac{18}{6}=3 \)
• Take another row, say \( x = 3 \), \( y = 9 \): \( \frac{9}{3}=3 \)

We notice that \( \frac{y}{x} \) is constant (equal to the constant of proportionality \( k \)) for all non - origin rows. This is the property of a proportional relationship: the ratio of \( y \) to \( x \) is constant (the slope \( k \)) for all points on the line (except the origin, where \( \frac{y}{x} \) is undefined in the sense of the ratio, but as \( x \) and \( y \) approach 0, the limit is \( k \)).

Answer:

For the table with \( x = 2,y = 6 \) and \( x = 6,y = 18 \), when we compute \( \frac{y}{x} \) for these rows and other non - origin rows, we find that \( \frac{y}{x} \) is constant (equal to 3 in this case). This shows that in a proportional relationship, the ratio of the \( y \)-value to the \( x \)-value (the constant of proportionality) is the same for all points on the line (excluding the origin, where the ratio is indeterminate but the limit is the constant of proportionality).