Question

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.)

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) \(y = 15x+103\)