Use the image to answer the question.

Question

An illustration shows a coordinate plane with 4 quadrants. The x-axis ranges from negative 9 to 9 in one unit increments, and the y-axis ranges from negative 11 to 11 in one unit increments. A line is graphed on the plane. An upward slanting line passes through points plotted at left parenthesis 1 comma 4 right parenthesis and left parenthesis 2 comma 8 right parenthesis.

Find the equation of the line y=mx through the origin that describes the proportional relationship shown in the graph.

(1 point)

Answer 1

To find the equation of the line \(y = mx\) that describes the proportional relationship shown in the graph, we first need to determine the slope \(m\) of the line that passes through the points \((1, 4)\) and \((2, 8)\).

The slope \(m\) is calculated using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the coordinates of the points:

• \((x_1, y_1) = (1, 4)\)
• \((x_2, y_2) = (2, 8)\)

we can calculate the slope as follows:

\[ m = \frac{8 - 4}{2 - 1} = \frac{4}{1} = 4 \]

Now that we have determined that the slope \(m\) is 4, we can write the equation of the line:

\[ y = 4x \]

Thus, the equation of the line that describes the proportional relationship shown in the graph is:

\[ \boxed{y = 4x} \]

Answer 2

Use the image to answer the question.

An illustration shows a graph labeled Football Cost. The horizontal axis is labeled Quantity and ranges from 0 to 5 in increments of 1. The vertical axis is labeled Cost and ranges from 0 to 39 in increments of 3. An upward slanting line segment connects points plotted at left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 9 right parenthesis, left parenthesis 2 comma 18 right parenthesis, left parenthesis 3 comma 27 right parenthesis, and left parenthesis 4 comma 36 right parenthesis.

The price for footballs is displayed in the graph. Derive the equation of the proportional relationship through the origin.

(1 point)

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Answer 3

To derive the equation of the proportional relationship through the origin based on the points given on the graph, we can use the points \((1, 9)\), \((2, 18)\), \((3, 27)\), and \((4, 36)\).

We will determine the slope \(m\) of the line connecting these points. The formula for slope is:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Let's calculate using the points \((1, 9)\) and \((2, 18)\):

\[ m = \frac{18 - 9}{2 - 1} = \frac{9}{1} = 9 \]

Now, we can write the equation of the line that represents the proportional relationship:

\[ y = mx \]

Substituting \(m = 9\) gives:

\[ y = 9x \]

Thus, the equation of the proportional relationship through the origin is:

\[ \boxed{y = 9x} \]

Answer 4

Use the image to answer the question.

An illustration shows a coordinate plane with 4 quadrants. The x-axis ranges from negative 9 to 9 in one unit increments, and the y-axis ranges from negative 11 to 11 in one unit increments. A line is graphed on the plane. An upward slanting line passes through points plotted at left parenthesis 3 comma 1 right parenthesis, left parenthesis 6 comma 2 right parenthesis, and left parenthesis 9 comma 3 right parenthesis.

Write the equation of the line y=mx through the origin that describes the proportional relationship shown in the graph.

(1 point)

Check answerRemaining Attempts : 3

Answer 5

To determine the equation of the line in the form \(y = mx\) that describes the proportional relationship shown in the graph, we first need to find the slope \(m\) of the line that passes through the points given: \((3, 1)\), \((6, 2)\), and \((9, 3)\).

The slope \(m\) can be calculated using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Let's take the points \((3, 1)\) and \((6, 2)\) as our reference points.

Calculating the slope \(m\):

\[ m = \frac{2 - 1}{6 - 3} = \frac{1}{3} \]

Now that we have the slope \(m = \frac{1}{3}\), we can write the equation of the line:

\[ y = \frac{1}{3}x \]

Therefore, the equation of the line that describes the proportional relationship through the origin is:

\[ \boxed{y = \frac{1}{3}x} \]