Question

b. does the graph show a proportional relationship? if so, what is the constant of proportionality ( k )? what is the equation of the graph? explain your reasoning.

Explanation:

To determine if a graph shows a proportional relationship, we use the following reasoning (assuming we had a graph, for example, a line through the origin with points like (1, 2), (2, 4), etc.):

Step 1: Check for Proportionality

A proportional relationship has a graph that is a straight line passing through the origin \((0,0)\). The equation for a proportional relationship is \(y = kx\), where \(k\) is the constant of proportionality.

Step 2: Find the Constant of Proportionality (\(k\))

If we have two points \((x_1, y_1)\) and \((x_2, y_2)\) on the line, \(k=\frac{y}{x}\) (since \(y = kx\) implies \(k=\frac{y}{x}\)). For example, if the graph passes through \((1, 3)\), then \(k=\frac{3}{1}=3\).

Step 3: Write the Equation

Using \(y = kx\), substitute \(k\) with the value found. If \(k = 3\), the equation is \(y = 3x\).

Since the graph isn't provided, here's a general approach:

1. Proportional Check: A proportional graph is a straight line through \((0,0)\).
2. Constant of Proportionality: \(k=\frac{y}{x}\) for any point \((x,y)\) on the line.
3. Equation: \(y = kx\) with the calculated \(k\).

For example, if the graph has points \((2, 6)\), then \(k=\frac{6}{2}=3\), so the equation is \(y = 3x\), and it is proportional.

If you provide the graph or specific points, we can calculate \(k\) and confirm the equation.

Answer:

To determine if a graph shows a proportional relationship, we use the following reasoning (assuming we had a graph, for example, a line through the origin with points like (1, 2), (2, 4), etc.):

Step 1: Check for Proportionality

A proportional relationship has a graph that is a straight line passing through the origin \((0,0)\). The equation for a proportional relationship is \(y = kx\), where \(k\) is the constant of proportionality.

Step 2: Find the Constant of Proportionality (\(k\))

If we have two points \((x_1, y_1)\) and \((x_2, y_2)\) on the line, \(k=\frac{y}{x}\) (since \(y = kx\) implies \(k=\frac{y}{x}\)). For example, if the graph passes through \((1, 3)\), then \(k=\frac{3}{1}=3\).

Step 3: Write the Equation

Using \(y = kx\), substitute \(k\) with the value found. If \(k = 3\), the equation is \(y = 3x\).

Since the graph isn't provided, here's a general approach:

1. Proportional Check: A proportional graph is a straight line through \((0,0)\).
2. Constant of Proportionality: \(k=\frac{y}{x}\) for any point \((x,y)\) on the line.
3. Equation: \(y = kx\) with the calculated \(k\).

For example, if the graph has points \((2, 6)\), then \(k=\frac{6}{2}=3\), so the equation is \(y = 3x\), and it is proportional.

If you provide the graph or specific points, we can calculate \(k\) and confirm the equation.