The equation f(x)= 2/3 x + 1 is giving me some hard times can anyone ?

F(x) = 2/3x + 1

This is in the slope-intercept form of an equation for a line. 
Which is written as this: (Note f(x)= is the same as saying y= in this case)

y = mx + b

M is our slope. Slope is the RISE over the RUN

In other words it is y/x
For this graph y represents distance, and x represents time. (Life Tip: TIME is almost ALWAYS the X value) 

In this case M = 2/3

So her speed is: 
2 units/3 units

B is the y-intercept. The y-intercept is where the X-value equals 0. So, (0,y). Because the y-intercept is distance she should have a y-intercept of 0. (The origin) because that would mean at the start of the race she had no distance (aka she was at the starting line).

Because her y-intercept (b) = 1 she did have a head start.

Here are both the answers at once to make it easier for you: 
Speed (Replace Units with whatever units you were given. Distance Units go on top. Time units on bottom. Example: Meters/Second))
2 units/3 units

Did she have a head start? (Unit being whatever distance unit you were given) 
Yes, 1 unit.