A one year old bluegill fish is 3 inches long and a four year old bluegill fish is 6 inches long.
Write an equation in slope-intercept form for the length, y, of a bluegill fish after x years.
To write the equation in slope-intercept form (y = mx + b), we need to determine the slope (m) and the y-intercept (b).
First, let's find the slope of the line that connects the two age-length points you've provided. The slope (m) is the change in the length of the fish divided by the change in the age of the fish, which can be calculated as follows:
m = (length at 4 years - length at 1 year) / (age at 4 years - age at 1 year)
m = (6 inches - 3 inches) / (4 years - 1 year)
m = 3 inches / 3 years
m = 1 inch/year
Now that we know the slope, we need to find the y-intercept (b), which is the length of the fish when the age (x) is zero. We can use either of the two points provided to figure out the y-intercept by rearranging the slope-intercept equation and solving for b. Let's use the point when the fish is 1 year old and 3 inches long:
3 inches = (1 inch/year) * (1 year) + b
3 inches = 1 inch + b
b = 3 inches - 1 inch
b = 2 inches
Therefore, the y-intercept (b) is 2 inches, and the slope (m) is 1 inch/year. The equation in slope-intercept form representing the length (y) of a bluegill fish after x years is:
y = mx + b
y = 1 inch/year * x years + 2 inches
y = x + 2
So, the final equation for the length of the bluegill fish in inches as a function of its age in years is:
y = x + 2