# Secant lines

## Function f

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## Function g

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## Instructions and Questions

1. Move the horizontal scroll bar to where x= -8. What is f(x)?

2. Where is f(x) = 0 ?

3. Is f relation a function? Why or Why not?

4. Move the horizontal scroll bar to where x= 2. What is g(x)?

5. Where is g(x) = 0 ?

6. Is g relation a function? Why or Why not?

7. Is there an inverse to either or both of these relations (Is it 1-1 "one-to one" or "injective")? Why or Why not?

8. Move the vertical scroll bar up to get a positive Δx ("delta-x" or "change in x"). Now move the horizontal scroll bar to see the line that connects these points (this is a secant line. Where is the secant line horizontal (zero-slope) for f?

9. Make the delta-x (Δx)=2 and move the horizontal bar to x=-8. What is the slope of the secant line? (use the information about f(x) and f(Δx) to compute the rise and run)

10. Check the instantaneous slope box. Is it the same?

11. Experiment with the Δx... what can you do to make the slope of the secant line look like it might match the instantaneous slope?

12. Is there any difference in the slope of the secant line if the delta-x is -2, as opposed to +2 for f?

13. Is there any difference in the slope of the secant line if the delta-x is -2, as opposed to +2 for g?

14. Move the x slider to find where the instantaneous slope is zero. List the values of x that make the slope zero for f.

15. Move the x slider to find where the instantaneous slope is zero. List the values of x that make the slope zero for g.

16. What do these places have in common? (Other than the fact the slope is zero)

17. What is the mathematical way of expressing f or g? (Hint: If you are not a functional analysis expert, see if you can find it in the source code of the applet! Check by typing it in your graphing calculator, changing the window size so that xMin=-16, xMax=+16 yMin -11 yMax=+11)