# Secant lines

## Function *f*

## Function *g*

## Instructions and Questions

- Move the horizontal scroll bar to where x= -8. What is f(x)?
- Where is f(x) = 0 ?
- Is
*f* relation a function? Why or Why not?
- Move the horizontal scroll bar to where x= 2. What is g(x)?
- Where is g(x) = 0 ?
- Is
*g* relation a function? Why or Why not?
- Is there an inverse to either or both of these relations (Is it 1-1 "one-to one" or "injective")? Why or Why not?
- 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*?
- 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)
- Check the instantaneous slope box. Is it the same?
- Experiment with the Δx... what can you do to make the slope of the secant line look
like it might match the instantaneous slope?
- Is there any difference in the slope of the secant line if the delta-x is -2, as opposed to +2 for
*f*?
- Is there any difference in the slope of the secant line if the delta-x is -2, as opposed to +2 for
*g*?
- Move the x slider to find where the instantaneous slope is zero. List the values of x that make
the slope zero for
*f*.
- Move the x slider to find where the instantaneous slope is zero. List the values of x that make
the slope zero for
*g*.
- What do these places have in common? (Other than the fact the slope is zero)
- 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*)*

## Links