t h(x) = x² - 8x.
(a) Find the average rate of change from 3 to 9.
(b) Find an equation of the secant line containing (3,h(3)) and (9, h(9))
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(a) The average rate of change from 3 to 9 can be found by calculating the slope of the secant line connecting the points (3, h(3)) and (9, h(9)).
The formula for the slope of a line passing through two points (x1, y1) and (x2, y2) is:
m = (y2 - y1) / (x2 - x1)
Applying this formula, we have:
m = (h(9) - h(3)) / (9 - 3)
To find the values of h(9) and h(3), we need to substitute 9 and 3 into the function h:
h(9) = 9² - 8(9) = -27 h(3) = 3² - 8(3) = -9
Substituting into the formula for slope, we have:
m = (-27 - (-9)) / (9 - 3) = -18 / 6 = -3
Therefore, the average rate of change from 3 to 9 is -3.
(b) To find the equation of the secant line containing (3, h(3)) and (9, h(9)), we need to find the slope of the line and its y-intercept.
We've already calculated the slope in part (a) as -3. To find the y-intercept, we can use the point-slope form of a line:
y - y1 = m(x - x1)
where m is the slope, (x1, y1) is a point on the line, and (x, y) are any other points on the line. We can choose either (3, h(3)) or (9, h(9)) as our point.
Let's use (3, h(3)):
y - h(3) = -3(x - 3)
Simplifying, we get:
y = -3x + 18 - 9 y = -3x + 9
Therefore, the equation of the secant line is y = -3x + 9.