Question

Which graph accurately depicts the correct formula for the points (1,5) , (2,7) , (3,9) , and (4,11) ?(1 point)
Responses

A A line passes through two points plotted on a coordinate plane.
Image with alt text: A line passes through two points plotted on a coordinate plane.

A A line passes through two points plotted on a coordinate plane.
Image with alt text: A line passes through two points plotted on a coordinate plane.

A A line passes through two points plotted on a coordinate plane.
Image with alt text: A line passes through two points plotted on a coordinate plane.

A A line passes through two points plotted on a coordinate plane.
Image with alt text: A line passes through two points plotted on a coordinate plane.

Answer 1

To determine which graph accurately depicts the relationship among the points (1,5), (2,7), (3,9), and (4,11), we should first identify the pattern in the coordinates.

Looking at the points:

- For \(x = 1\), \(y = 5\)
- For \(x = 2\), \(y = 7\)
- For \(x = 3\), \(y = 9\)
- For \(x = 4\), \(y = 11\)

We can observe that as \(x\) increases by 1, \(y\) increases by 2. This suggests a linear relationship.

To find the equation of the line that best describes this relationship, we can set it in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

1. Calculate the slope (\(m\)):
- Change in \(y\) = 7 - 5 = 2 (from point (1,5) to (2,7))
- Change in \(x\) = 2 - 1 = 1
- Therefore, \(m = \frac{2}{1} = 2\).

2. Using the slope-intercept form and one of the points to find \(b\):
- Take point (1,5):
\[
5 = 2(1) + b \implies 5 = 2 + b \implies b = 3.
\]

Thus, the equation of the line is \(y = 2x + 3\).

Now, we need to check if this line passes through all the points:

- Substituting \(x = 1\): \(y = 2(1) + 3 = 5\) → point (1,5).
- Substituting \(x = 2\): \(y = 2(2) + 3 = 7\) → point (2,7).
- Substituting \(x = 3\): \(y = 2(3) + 3 = 9\) → point (3,9).
- Substituting \(x = 4\): \(y = 2(4) + 3 = 11\) → point (4,11).

All points satisfy the equation \(y = 2x + 3\).

Since I don't have access to the images of the graphs, look for a graph that depicts a linear function with a slope of 2 that passes through the points mentioned. It should have a y-intercept of 3 and should ideally show a straight line going through points like (1, 5), (2, 7), (3, 9), and (4, 11). The correct graph will represent this linear relationship.