Function f has constant first differences. Which of the following must be true?

A) The graph of f is a parabola.
B) The function f has constant values.
C) The second differences for f are all zero.
D) The growth rate for f increases as x increases

C) The second difference for f are all zero.

Step-by-step explanation:

Hi there!

1) The graph of f is a parabola.

If Function f had constant second differences, then the graph would be a parabola. However, because the first differences are constant, this indicates that the graph of f is a line.

B) The function f has constant values.

This option states that Function f must have constant values in its equation. We don't know this as we are not given an equation and we can't determine this solely on the fact that the first differences are constant.

The equation could be f(x)=x, f(x)=2x+1 or any linear equation.

C) The second differences for f are all zero.

This makes sense since the first differences are all constant.

If the first differences were all 5, for example, 5-5=0 and that would occur for every second difference.

D) The growth rate for f increases as x increases.

It is also completely possible for the value of f(x) to decrease as x increases.

If the first differences were positive, f would increase as x increases.

If they were negative, f would decrease as x increases.

We're only given that the first differences are constant, so again, we don't know this.

I hope this helps!