The rate of change, the initial value and the equation of the function are;
Rate of change = -110 m/min (the speed with which Penelope walks)
Initial value = 2,480 m (Penelope's initial distance from the park at time t = 0 minutes)
The equation of the function is; y = 2,480 - 110·x
The above values an equation are found as follows;
The first difference of the values are; 1380 - 1930 = -550
830 - 1380 = -550
280 - 830 = -550
The first difference is constant, therefore, the relationship is a linear relationship
The slope, m, of the graph of the function is the rate of change, which can be found as follows;
Slope, m is; -550/(10 - 5) = -110
The rate of change is -110, which indicates that Penelope's distance from the park reduces by 110 meters every 5 minutes, which is Penelope's walking speed
The initial value is Penelope's initial distance when she started walking to the park, which is at the time, t = 0 minutes; We get that 0 = 5 - 5, therefore, the initial value is the distance value 5 minutes before Penelope's distance from the park was 1,930
The rate of change of -110 meters per minute, indicates that Penelope's distance from the park at time 0 minutes is 1930 + 5 × 110 = 2,480
The initial value is; 2,480
The slope of the function is; -110, and the initial value, which is the y-intercept is 2,480. The equation of the function in slope intercept form is therefore;
y = 2,480 - 110·x