Question

card 5
refer to the graph.
graph of line m passing through the origin with a positive slope
a draw a line that is a vertical translation of line m.
b write equations for both lines.
c which of these lines represents a proportional relationship? explain your thinking.

Explanation:

Step1: Analyze Line m

Line \( m \) passes through the origin \((0,0)\) and has a slope. Let's find the slope using two points. For example, when \( x = 6 \), \( y = 4 \), so slope \( m=\frac{4 - 0}{6 - 0}=\frac{2}{3}\). So equation of line \( m \) is \( y=\frac{2}{3}x \) (since it passes through \((0,0)\), \( b = 0 \)).

Step2: Vertical Translation

A vertical translation of a line \( y = mx + b \) is \( y = mx + b + k \) (where \( k
eq0 \) is the translation amount). Let's choose \( k = 3 \), so the translated line will be \( y=\frac{2}{3}x+3 \). To draw it, we can use the slope and y - intercept. The y - intercept is \( 3 \), so plot \((0,3)\) and then use the slope \( \frac{2}{3} \) (rise 2, run 3) to find other points.

Step3: Equations for Both Lines

• Line \( m \): As found, \( y=\frac{2}{3}x \) (since it passes through the origin, \( b = 0 \)).
• Translated line (let's say we translated up by 3 units): \( y=\frac{2}{3}x + 3 \)

Step4: Proportional Relationship

A proportional relationship is of the form \( y = kx \) (where \( k \) is a constant), which means the line must pass through the origin \((0,0)\) (since when \( x = 0 \), \( y = 0 \)). Line \( m \) passes through \((0,0)\) and has the equation \( y=\frac{2}{3}x \), so it represents a proportional relationship. The translated line \( y=\frac{2}{3}x + 3 \) does not pass through the origin (when \( x = 0 \), \( y = 3
eq0 \)), so it is not a proportional relationship.

Answer:

a) (Drawing: A line parallel to \( m \) shifted up or down, e.g., \( y=\frac{2}{3}x + 3 \) plotted with y - intercept 3 and same slope as \( m \))
b) Line \( m \): \( y=\frac{2}{3}x \); Translated line (e.g., up by 3): \( y=\frac{2}{3}x + 3 \)
c) Line \( m \) represents a proportional relationship because its equation is of the form \( y = kx \) (passes through the origin \((0,0)\)) and \( y\) is directly proportional to \( x \) with constant of proportionality \( \frac{2}{3} \). The translated line has a non - zero y - intercept, so it is not of the form \( y = kx \).