the function ( y = 20x^{2}+12x - 8 ) is graphed in the standard ( (x,y) ) coordinate plane. through which of…

Explanation:

Step1: Find the derivative of the function

The function is (y = 20x^{2}+12x - 8). Using the power rule ((x^{n})^\prime=nx^{n - 1}), the derivative (y^\prime=40x + 12).

Step2: Find the critical point

Set (y^\prime = 0), so (40x+12 = 0). Solving for (x): [ \begin{align*} 40x&=- 12\ x&=-\frac{12}{40}=-\frac{3}{10}=-0.3 \end{align*} ] For a quadratic function (y = ax^{2}+bx + c) ((a = 20>0)), the function is increasing when (y^\prime>0). Solving (40x + 12>0) gives (x>-\frac{3}{10}).

Step3: Check each interval

• For interval (A=(1,3)): Since (1>-\frac{3}{10}), for all (x\in(1,3)), (y^\prime = 40x + 12>0). The function is increasing on ((1,3)).
• For interval (C=[-1,3]): When (x=-1), (y^\prime=40\times(-1)+12=-28<0). The function is not increasing at (x = - 1) in the interval ([-1,3]).

Answer:

A. ((1,3))