Question
There is a linear relationship between the number of roses blooming in the garden and the number of days that pass. After 4 days, there are 22 rose blooms left, and after 8 days, there are 10 rose blooms left.
Which of the following graphs represents the relationship?
a coordinate plane with the x-axis labeled time in days and the y-axis labeled number of roses blooming, with a line segment that passes through the points 0 comma 34 and 8 comma 10
a coordinate plane with the x-axis labeled time in days and the y-axis labeled number of roses blooming, with a line segment that passes through the points 0 comma 36 and 8 comma 12
a coordinate plane with the x-axis labeled time in days and the y-axis labeled number of roses blooming, with a line segment that passes through the points 0 comma 38 and 8 comma 14
a coordinate plane with the x-axis labeled time in days and the y-axis labeled number of roses blooming, with a line segment that passes through the points 0 comma 40 and 8 comma 16
Which of the following graphs represents the relationship?
a coordinate plane with the x-axis labeled time in days and the y-axis labeled number of roses blooming, with a line segment that passes through the points 0 comma 34 and 8 comma 10
a coordinate plane with the x-axis labeled time in days and the y-axis labeled number of roses blooming, with a line segment that passes through the points 0 comma 36 and 8 comma 12
a coordinate plane with the x-axis labeled time in days and the y-axis labeled number of roses blooming, with a line segment that passes through the points 0 comma 38 and 8 comma 14
a coordinate plane with the x-axis labeled time in days and the y-axis labeled number of roses blooming, with a line segment that passes through the points 0 comma 40 and 8 comma 16
Answers
Answered by
GPT-4o mini
To determine the correct graph that represents the linear relationship between the number of roses blooming and the number of days that pass, we can first find the equation of the line using the given points:
1. We have two points:
- \( (4, 22) \) and \( (8, 10) \)
2. To find the slope \( m \) of the line, we use the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{10 - 22}{8 - 4} = \frac{-12}{4} = -3
\]
3. Now that we have the slope \( m = -3 \), we can use one of the points to find the equation of the line. We'll use the point \( (4, 22) \) and the point-slope form of the line:
\[
y - y_1 = m(x - x_1)
\]
Plugging in the values gives us:
\[
y - 22 = -3(x - 4)
\]
Simplifying this, we get:
\[
y - 22 = -3x + 12 \\
y = -3x + 34
\]
4. Now, to find the y-intercept (when \( x = 0 \)), we substitute \( x = 0 \):
\[
y = -3(0) + 34 = 34
\]
Thus, the equation of the line is \( y = -3x + 34 \), which indicates that when \( x = 0 \) (0 days), there are 34 rose blooms.
To check the value after 8 days:
\[
y = -3(8) + 34 = -24 + 34 = 10
\]
This confirms that after 8 days, there are indeed 10 blooms left.
Now, referring to the answer options:
- The first option states that the line passes through \( (0, 34) \) and \( (8, 10) \), which matches our findings.
- None of the other options match the intercept or the derived points.
Therefore, the correct graph that represents the relationship is the first one:
**a coordinate plane with the x-axis labeled time in days and the y-axis labeled number of roses blooming, with a line segment that passes through the points \( (0, 34) \) and \( (8, 10) \)**.
1. We have two points:
- \( (4, 22) \) and \( (8, 10) \)
2. To find the slope \( m \) of the line, we use the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{10 - 22}{8 - 4} = \frac{-12}{4} = -3
\]
3. Now that we have the slope \( m = -3 \), we can use one of the points to find the equation of the line. We'll use the point \( (4, 22) \) and the point-slope form of the line:
\[
y - y_1 = m(x - x_1)
\]
Plugging in the values gives us:
\[
y - 22 = -3(x - 4)
\]
Simplifying this, we get:
\[
y - 22 = -3x + 12 \\
y = -3x + 34
\]
4. Now, to find the y-intercept (when \( x = 0 \)), we substitute \( x = 0 \):
\[
y = -3(0) + 34 = 34
\]
Thus, the equation of the line is \( y = -3x + 34 \), which indicates that when \( x = 0 \) (0 days), there are 34 rose blooms.
To check the value after 8 days:
\[
y = -3(8) + 34 = -24 + 34 = 10
\]
This confirms that after 8 days, there are indeed 10 blooms left.
Now, referring to the answer options:
- The first option states that the line passes through \( (0, 34) \) and \( (8, 10) \), which matches our findings.
- None of the other options match the intercept or the derived points.
Therefore, the correct graph that represents the relationship is the first one:
**a coordinate plane with the x-axis labeled time in days and the y-axis labeled number of roses blooming, with a line segment that passes through the points \( (0, 34) \) and \( (8, 10) \)**.