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), where y is the length of the bluegill fish, x is the age in years, m is the slope (rate of growth per year), and b is the y-intercept (the size of the fish at year zero), we need to calculate the slope using the information given.
We are given two points that represent the age and length of a bluegill fish at those ages:
Point 1 (x1, y1) = (1, 3) (at one year old, the fish is 3 inches long)
Point 2 (x2, y2) = (4, 6) (at four years old, the fish is 6 inches long)
First, let's find the slope (m) using the two points:
m = (y2 - y1) / (x2 - x1)
Substituting the points into the equation:
m = (6 - 3) / (4 - 1)
m = 3 / 3
m = 1
So, the slope is 1 inch per year.
Now, we need to find the y-intercept (b). We already have the slope, so we can use one of the points to find b. Let's use the first point (1, 3).
We know that:
y = mx + b
So substituting the values from the first point, and the slope we just found:
3 = (1)(1) + b
Now solve for b:
3 = 1 + b
b = 3 - 1
b = 2
Now we have both m and b, so we can write the equation in slope-intercept form:
y = 1x + 2
Or simply:
y = x + 2
Thus, the equation for the length (y) of a bluegill fish after x years is y = x + 2.