Using Trend Lines Quick Check
Use the image to answer the question.
An illustration shows the first quadrant of a coordinate plane titled Population of a City Since 1900.
The x-axis shows years since 1900 and ranges from 0 to 80 in increments of 10. The y-axis shows population per thousand and ranges from 0 to 80 in increments of 5. Nine points are plotted on the graph. The points are plotted at approximate coordinates left parenthesis 0 comma 20 right parenthesis, left parenthesis 10 comma 28 right parenthesis, left parenthesis 20 comma 30 right parenthesis, left parenthesis 30 comma 35 right parenthesis, left parenthesis 40 comma 47 right parenthesis, left parenthesis 50 comma 55 right parenthesis, left parenthesis 60 comma 57 right parenthesis, left parenthesis 70 comma 68 right parenthesis, and left parenthesis 80 comma 66 right parenthesis. An upward slanting line starts at left parenthesis 0 comma 20 right parenthesis and continues to left parenthesis 60 comma 57 right parenthesis.
The population growth of a city since 1900 is represented by a linear model. Interpret the slope and the y -intercept.
For every year since 1900, the population grew by approximately 650. In 1900, the population was 20,000.
For every year since 1900, the population grew by approximately 650. In 1900, the population was 20,000.
For every year since 1900, the population grew by approximately 300. In 1900, the population was 0.
For every year since 1900, the population grew by approximately 300. In 1900, the population was 0.
For every year since 1900, the population grew by approximately 1,590. in 1900, the population was 20,000.
For every year since 1900, the population grew by approximately 1,590. in 1900, the population was 20,000.
For every 0.65 of a year since 1900, the population grew by approximately 1,000. In 1900, the population was 20,000.
For every year since 1900, the population grew by approximately 650. In 1900, the population was 20,000.
The correct interpretation is:
For every year since 1900, the population grew by approximately 650. In 1900, the population was 20,000.
To interpret the slope and y-intercept of the linear model representing the population growth of a city since 1900, we need to understand the equation of a straight line: y = mx + b.
In the given context:
- The slope (m) represents the rate of change of the population per year.
- The y-intercept (b) represents the initial population in the year 1900.
Looking at the image and the data points provided, we can identify the following details:
1. The upward slanting line starts at (0, 20) and continues to (60, 57).
2. The line represents the growth of the population since 1900.
3. The x-axis represents the years since 1900.
4. The y-axis represents the population per thousand.
From the given information, we can conclude:
- The slope of the line represents the rate of change of the population per year.
- By observing the points on the line, we can approximate the slope as (57 - 20)/(60 - 0) = 37/60 ≈ 0.617.
- This means that for every year since 1900, the population grew by approximately 0.617 per year.
To determine the y-intercept:
- We can see that the line intersects the y-axis at the point (0, 20).
- This means that in the year 1900, the population was 20,000.
Taking all of this into account, the correct interpretation is:
"For every year since 1900, the population grew by approximately 0.617. In 1900, the population was 20,000."