let f(x)=x^2 +2x. find the slope of the secant line joining (1,f(1) and (8,f(8).

We are asked to find the slope of the secant line between two points on the function \( f(x) = x^2 + 2x \).

### Step 1: Calculate \( f(1) \)
First, substitute \( x = 1 \) into \( f(x) \):
\[
f(1) = 1^2 + 2(1) = 1 + 2 = 3
\]
So, the point is \( (1, f(1)) = (1, 3) \).

### Step 2: Calculate \( f(8) \)
Now, substitute \( x = 8 \) into \( f(x) \):
\[
f(8) = 8^2 + 2(8) = 64 + 16 = 80
\]
So, the point is \( (8, f(8)) = (8, 80) \).

### Step 3: Use the slope formula
The slope \( m \) of a secant line between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substitute the values \( (1, 3) \) and \( (8, 80) \):
\[
m = \frac{80 - 3}{8 - 1} = \frac{77}{7} = 11
\]

### Final Answer:
The slope of the secant line is \( 11 \).

Would you like more details on the calculation process, or do you have any other questions?

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Here are 5 related questions to explore:

1. What is the equation of the secant line between \( (1, f(1)) \) and \( (8, f(8)) \)?
2. How do you find the derivative of \( f(x) = x^2 + 2x \)?
3. What is the difference between a secant line and a tangent line?
4. Can you calculate the slope of the secant line for a different interval, say between \( x = 3 \) and \( x = 7 \)?
5. What is the instantaneous rate of change of \( f(x) \) at \( x = 1 \)?

**Tip**: The slope of a secant line gives the average rate of change of the function over the interval!

let f(x)=x^2 +2x. find the slope of the secant line joining (1,f(1)) and (8,f(8)).

Learn how to find the slope of the secant line for the quadratic function f(x) = x^2 + 2x between the points (1, f(1)) and (8, f(8)). This step-by-step solution involves calculating the values of f(x) at the given points and applying the slope formula, suitable for students in Grades 9-11.

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