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Let g(x)=x²-1.

(a) Find the average rate of change from 2 to 6.
(b) Find an equation of the secant line containing (-2, g(-2)) and (6, g(6)).

User Maxim Kim
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1 Answer

18 votes
18 votes

Answer:

Explanation:

To find the average rate of change from 2 to 6, we can use the formula for the average rate of change, which is given by (y2 - y1) / (x2 - x1). In this case, the function we are working with is g(x) = x² - 1, so we need to plug in the values of x1 and x2 into the function to get the corresponding values of y1 and y2.

a) The average rate of change from 2 to 6 is given by:

(g(6) - g(2)) / (6 - 2)

Plugging in the values of x1 and x2 into the function g(x), we get:

(g(6) - g(2)) / (6 - 2) = (6² - 1 - 2² + 1) / (6 - 2) = (35 - 3) / 4 = 32 / 4 = 8

Therefore, the average rate of change from 2 to 6 is 8.

b) To find an equation of the secant line containing the points (-2, g(-2)) and (6, g(6)), we need to first find the slope of the secant line. The slope of the secant line is given by the formula (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two points on the secant line.

In this case, the coordinates of the two points on the secant line are (-2, g(-2)) and (6, g(6)). Plugging these values into the formula for the slope, we get:

m = (g(6) - g(-2)) / (6 - (-2)) = (6² - 1 - (-2)² + 1) / (6 + 2) = (35 + 5) / 8 = 40 / 8 = 5

Now that we know the slope of the secant line, we can use the point-slope form of a line to write the equation of the secant line. The point-slope form of a line is given by the formula y - y1 = m(x - x1), where (x1, y1) is a point on the line, m is the slope of the line, and (x, y) are the coordinates of a generic point on the line.

In this case, we know that the point (-2, g(-2)) is on the secant line, and we know that the slope of the secant line is 5. Plugging these values into the point-slope form of a line, we get:

y - g(-2) = 5(x - (-2))

We can then simplify this equation to get:

y - (-2² - 1) = 5(x + 2)

y + 3 = 5x + 10

Finally, we can move all of the terms containing x to one side of the equation and all of the constant terms to the other side of the equation to get the standard form of the equation of the line:

y = 5x + 7

Therefore, an equation of the secant line containing the points (-2, g(-2)) and (6, g(6)) is y = 5x + 7.

User Moreno
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