The points that the graph line passes through on the graph are (- 1, - 4), (0, 1), (1, 4) and (2, 11)
The equation of the polynomial can be expressed in the slope intercept form which is expressed as
y = mx + c
where
m represents slope
c represents y intercept
The formula for determining slope is expressed as
m = (y2 - y1)/(x2 - x1)
Considering the first two points,
x1 = - 1, y1 = - 4
x2 = 0, y2 = 1
m = (1 - - 4)/(0 - - 1) = (1 + 4)/(0 + 1) = 5/1
m = 5
We would find the y intercept by substituting m = 5, x = - 1 and y = - 4 into the equation. It becomes
- 4 = 5 * - 1 + c
- 4 = - 5 + c
c = - 4 + 5
c = 1
By substituting m = 5 and c = 1 into the slope intercept equation, the equation of the line would be
y = 5x + 1
We can check by substituting x = 2 and y = 11 from the last point into the equation. It becomes
11 = 5 * 2 + 1
11 = 11
To express it as a function, we would replace y with f(x). Thus, the function whose graph passes through (-1,-4) (0,1),(1,4) and (2,11) is
f(x) = 5x + 1