Question

each graph below shows a relationship between x and y.
for each graph, determine whether x and y are proportional.
if x and y are proportional, fill in the blank with a number.
graph 1
graph of a line passing through (0,0), (1,1), (2,2)
\bigcirc proportional
\quad y is \square times x
\bigcirc not proportional
graph 2
graph of a line passing through (0,0), (1,3), (2,6)
\bigcirc proportional
\quad y is \square times x
\bigcirc not proportional

Explanation:

Graph 1

Step1: Check proportionality

A proportional relationship has a graph that is a straight line through the origin, and \( y = kx \) (where \( k \) is constant). Graph 1 passes through \((0,0)\) and points like \((1,1)\), \((2,2)\).

Step2: Find the constant \( k \)

Using the point \((x = 1, y = 1)\), from \( y=kx \), we get \( 1 = k\times1 \), so \( k = 1 \). Also, \((2,2)\): \( 2 = k\times2 \), \( k = 1 \). So \( y = 1\times x \).

Step1: Check proportionality

Graph 2 passes through \((0,0)\) and points like \((1,3)\), \((2,6)\). A straight line through origin, so check \( k \).

Step2: Find the constant \( k \)

For \((x = 1, y = 3)\), \( y=kx \) gives \( 3 = k\times1 \), \( k = 3 \). For \((2,6)\): \( 6 = k\times2 \), \( k = 3 \). So \( y = 3\times x \).

Answer:

Proportional, \( y \) is \( 1 \) times \( x \)

Graph 2