The point P(5, 1) lies on the curve 
y = 

x − 4
.
(a) If Q is the point 
(x, 

x − 4
),
 use your calculator to find the slope of the secant line PQ (correct to six decimal places) for the following values of x.
(i)    4.5


(ii)    4.9


(iii)    4.99


(iv)    4.999


(v)    5.5


(vi)    5.1


(vii)    5.01
4
 
Incorrect: Your answer is incorrect.
 seenKey 0.498756

(viii)    5.001


To find the slope of the secant line between the points \( P(5, 1) \) and \( Q(x, x-4) \), we use the formula for the slope of a secant line:

\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Here, \( P(5, 1) \) is one point, and \( Q(x, x-4) \) is the other point. So, the slope between these two points is:

\[
\text{slope} = \frac{(x - 4) - 1}{x - 5} = \frac{x - 4 - 1}{x - 5} = \frac{x - 5}{x - 5}
\]

Now we can calculate the slope for each given value of \( x \).

### (i) When \( x = 4.5 \):

\[
\text{slope} = \frac{4.5 - 5}{4.5 - 5} = \frac{-0.5}{-0.5} = 1
\]

### (ii) When \( x = 4.9 \):

\[
\text{slope} = \frac{4.9 - 5}{4.9 - 5} = \frac{-0.1}{-0.1} = 1
\]

### (iii) When \( x = 4.99 \):

\[
\text{slope} = \frac{4.99 - 5}{4.99 - 5} = \frac{-0.01}{-0.01} = 1
\]

### (iv) When \( x = 4.999 \):

\[
\text{slope} = \frac{4.999 - 5}{4.999 - 5} = \frac{-0.001}{-0.001} = 1
\]

### (v) When \( x = 5.5 \):

\[
\text{slope} = \frac{5.5 - 5}{5.5 - 5} = \frac{0.5}{0.5} = 1
\]

### (vi) When \( x = 5.1 \):

\[
\text{slope} = \frac{5.1 - 5}{5.1 - 5} = \frac{0.1}{0.1} = 1
\]

### (vii) When \( x = 5.01 \):

\[
\text{slope} = \frac{5.01 - 5}{5.01 - 5} = \frac{0.01}{0.01} = 1
\]

### (viii) When \( x = 5.001 \):

\[
\text{slope} = \frac{5.001 - 5}{5.001 - 5} = \frac{0.001}{0.001} = 1
\]

As you can see, the slope of the secant line PQ is consistently 1 for all values of \( x \), regardless of how close or far \( x \) is from 5.

---

Do you want more details on how the slope was calculated? Here are 5 related questions you could explore:

1. How does the slope of the secant line relate to the derivative of the function?
2. What happens to the secant line as \( x \) gets infinitely close to 5?
3. How would you calculate the tangent line at \( x = 5 \)?
4. What if the curve were different, how would the slope calculation change?
5. Can you find the equation of the secant line for any general point \( Q(x, x-4) \)?

**Tip**: The slope of a secant line approaches the derivative of the function as the two points get closer!

The point P(5, 1) lies on the curve y = x − 4. (a) If Q is the point (x, x − 4), use your calculator to find the slope of the secant line PQ (correct to six decimal places) for the following values of x. (i) 4.5 (ii) 4.9 (iii) 4.99 (iv) 4.999 (v) 5.5 (vi) 5.1 (vii) 5.01 (viii) 5.001

Explore the process of calculating the slope of a secant line for point P(5,1) on the curve y = x - 4 using different values of x. Learn about the slope formula and how limits approach the derivative. Suitable for students in Grades 10-12.

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