Given the function f(x) = 3x - 9

a. Find the rate of change between the two stated values for x: 7 to 8

b. Find the equation of a secant line containing the given points: (7, f(7)) and (8, f(8))

curious question in Calculus.

Since the function is a straight line, it seems very odd to be talking about secant lines, anyway ....

f(7) = 12
f(8) = 15
rate of change = (15-12)/(8-7) = 3

Note that this is the slope of the given line as expected.
The equation of the secant line would of course be the line itself , or f(x) = 3x-9

To find the rate of change between two values for x, we need to calculate the difference in the function values divided by the difference in the x-values. In this case, we want to find the rate of change between x = 7 and x = 8.

a. Rate of change between x = 7 and x = 8:
To find the rate of change between two values, we subtract the initial value from the final value and divide by the change in x. In this case, we have f(7) and f(8) as the initial and final values respectively.

Step 1: Substitute x = 7 into the function f(x) = 3x - 9
f(7) = 3(7) - 9
f(7) = 21 - 9
f(7) = 12

Step 2: Substitute x = 8 into the function f(x) = 3x - 9
f(8) = 3(8) - 9
f(8) = 24 - 9
f(8) = 15

Step 3: Calculate the rate of change
Rate of change = (f(8) - f(7))/(8 - 7)
Rate of change = (15 - 12)/(8 - 7)
Rate of change = 3/1
Rate of change = 3

Therefore, the rate of change between x = 7 and x = 8 in the function f(x) = 3x - 9 is 3.

b. Equation of the secant line:
To find the equation of the secant line passing through two given points, we need to calculate the slope (rate of change) and then use the point-slope formula.

Given points: (7, f(7)) and (8, f(8))
We already found the function values in part a.

Point 1: (7, 12)
Point 2: (8, 15)

Step 1: Calculate the slope (rate of change)
Slope = (y2 - y1)/(x2 - x1)
Slope = (15 - 12)/(8 - 7)
Slope = 3/1
Slope = 3

Step 2: Use the point-slope formula to find the equation of the line
y - y1 = m(x - x1) (where m is the slope, and (x1, y1) is one of the given points)

Using point 1: (7, 12)
y - 12 = 3(x - 7)

Simplifying, we get:
y - 12 = 3x - 21

Rearranging to the slope-intercept form:
y = 3x - 21 + 12
y = 3x - 9

Therefore, the equation of the secant line passing through the points (7, f(7)) and (8, f(8)) is y = 3x - 9.