let g(x)=3x^2 - 5. (a) find the average rate of change from - 4 to 9. (b) find an equation of the secant…

let g(x)=3x^2 - 5. (a) find the average rate of change from - 4 to 9. (b) find an equation of the secant line containing (-4,g(-4)) and (9,g(9)). (a) the average rate of change from - 4 to 9 is . (simplify your answer.) (b) an equation of the secant line containing (-4,g(-4)) and (9,g(9)) is . (type your answer in slope - intercept form.)

let g(x)=3x^2 - 5. (a) find the average rate of change from - 4 to 9. (b) find an equation of the secant line containing (-4,g(-4)) and (9,g(9)). (a) the average rate of change from - 4 to 9 is . (simplify your answer.) (b) an equation of the secant line containing (-4,g(-4)) and (9,g(9)) is . (type your answer in slope - intercept form.)

Answer

Explanation:

Step1: Find g(-4) and g(9)

Given (g(x)=3x^{2}-5), then (g(-4)=3\times(-4)^{2}-5=3\times16 - 5=48 - 5 = 43) and (g(9)=3\times9^{2}-5=3\times81-5 = 243-5=238).

Step2: Calculate the average rate of change

The formula for the average rate of change of a function (y = g(x)) from (x=a) to (x = b) is (\frac{g(b)-g(a)}{b - a}). Here (a=-4) and (b = 9), so the average rate of change is (\frac{g(9)-g(-4)}{9-(-4)}=\frac{238 - 43}{9 + 4}=\frac{195}{13}=15).

Step3: Find the equation of the secant - line

The slope - intercept form of a line is (y=mx + c), where (m) is the slope and (c) is the y - intercept. We know the slope (m = 15), and we can use the point ((-4,43)) to find (c). Substitute (x=-4), (y = 43) and (m = 15) into (y=mx + c): (43=15\times(-4)+c), (43=-60 + c), then (c=43 + 60=103). So the equation of the secant line is (y = 15x+103).

Answer:

(a) 15 (b) (y = 15x+103)