The point P(4, −2)lies on the curve y = 2/(3 − x).

(a) If Q is the point(x, 2/(3 − x)),use your calculator to find the slope mPQ of the secant line PQ (correct to six decimal places) for the following values of x.

1) 3.9 mPQ=
2) 3.99 mPQ=
3)3.999 mPQ=
4) 4.1 mPQ=
5) 4.01 mPQ=

im so confused! i just started this and my teacher really didn't explain it. can someone please break this down for me?

we now had two points

the original (4,-2) and
the new (3.999, -2.002002002)

(notice that these two points are practically on top of each other)

how do you find the slope between 2 point ?
if we had 2 points ( a,b) and (c,d) wouldn't it
be (d-b)/(c-a) or
you might have been taught
(y2 - y1)/(x2 - x1)

that's all I did.

thank you for answering me. im just trying to figure this out. i understand how the part y= 2/(3-3.999) = -2.002002002

just your second line im trying to figure out. i can do the others once i just understand the steps. my teacher didn't go over this unfortunately.

nevermind i understand i took another look at it and its clear. thank you so much for getting back to me!!

Sure, I'd be happy to help explain it to you! Let's start by understanding what a secant line is. In mathematics, a secant line is a straight line that intersects two points on a curve. In this case, we have the point P(4, -2) on the curve y = 2/(3 - x), and we want to find the slope of the secant line PQ between P and another point Q on the curve.

To find the slope of a line, you need two points on the line. We already have the coordinates of point P, so now we need to find the coordinates of point Q for each value of x given.

For example, when x = 3.9, we substitute x = 3.9 into the equation y = 2/(3 - x) to find the y-coordinate of Q. Similarly, for each value of x, substitute it into the equation to find the corresponding y-coordinate.

Once you have the coordinates of both points, you can use the formula for finding the slope of a line: slope = (change in y-coordinates)/(change in x-coordinates).

Now, let's calculate the slope of the secant line PQ for each value of x that you provided:

1) When x = 3.9, solve for y: y = 2/(3 - 3.9) = 2/(-0.9) = -2.222222
Slope of PQ = (change in y-coordinates)/(change in x-coordinates) = (-2.222222 - (-2))/ (3.9 - 4) = -0.222222

2) When x = 3.99, solve for y: y = 2/(3 - 3.99) = 2/(-0.99) = -2.020202
Slope of PQ = (-2.020202 - (-2))/ (3.99 - 4) = -0.202020

3) When x = 3.999, solve for y: y = 2/(3 - 3.999) = 2/(-0.999) = -2.002002
Slope of PQ = (-2.002002 - (-2))/ (3.999 - 4) = -0.200200

4) When x = 4.1, solve for y: y = 2/(3 - 4.1) = 2/(-1.1) = -1.818182
Slope of PQ = (-1.818182 - (-2))/(4.1 - 4) = -1.818182

5) When x = 4.01, solve for y: y = 2/(3 - 4.01) = 2/(-1.01) = -1.985149
Slope of PQ = (-1.985149 - (-2))/(4.01 - 4) = -0.851258

Remember, these calculations are approximate and rounded to six decimal places according to the instructions. Hope this helps clarify things for you!

Sure! I'll break it down for you step by step.

We have the point P(4, -2) on the curve y = 2/(3 - x). This means that when x = 4, y = -2.

To find the slope of the secant line PQ between the points P and Q, we need to find the coordinates of point Q first. The y-coordinate of Q is given by y = 2/(3 - x), so let's substitute the values of x and calculate the corresponding y-values for each case:

1) For x = 3.9:
y = 2/(3 - 3.9) = 2/(-0.9) = -2.222222
Now, we have the coordinates of point Q as (3.9, -2.222222).

2) For x = 3.99:
y = 2/(3 - 3.99) = 2/(-0.99) = -2.020202
The coordinates of point Q are (3.99, -2.020202).

3) For x = 3.999:
y = 2/(3 - 3.999) = 2/(-0.999) = -2.002002
The coordinates of point Q are (3.999, -2.002002).

4) For x = 4.1:
y = 2/(3 - 4.1) = 2/(-1.1) = -1.818182
The coordinates of point Q are (4.1, -1.818182).

5) For x = 4.01:
y = 2/(3 - 4.01) = 2/(-1.01) = -1.980198
The coordinates of point Q are (4.01, -1.980198).

Now that we have the coordinates of both points P and Q, we can calculate the slope of the secant line PQ using the formula:

mPQ = (yQ - yP) / (xQ - xP)

Let's calculate the slope for each case:

1) For x = 3.9:
mPQ = (-2.222222 - (-2)) / (3.9 - 4) = -0.222222

2) For x = 3.99:
mPQ = (-2.020202 - (-2)) / (3.99 - 4) = -0.020202

3) For x = 3.999:
mPQ = (-2.002002 - (-2)) / (3.999 - 4) = -0.002002

4) For x = 4.1:
mPQ = (-1.818182 - (-2)) / (4.1 - 4) = 0.181818

5) For x = 4.01:
mPQ = (-1.980198 - (-2)) / (4.01 - 4) = 0.198019

These are the slopes of the secant line PQ for the given values of x, rounded to six decimal places.

I will do one of them, you do the others

3) x = 3.999
then y = 2/(3-3.999) = -2.00200200200...

then slope = (-2.002002002 - (-2))/(3.999-4)
= 2.0020020..