Question

1. (5 points) graph the function: (g(x) = 3|x| - 1)

1. (10 points) calculate the difference quotient (dq) of the given function. simplify your answer fully.

(f(x) = 3x^2 - 4)

(dq =)

Explanation:

Step1: Identify vertex of the absolute value function

$$(0, -1)$$

Step2: Determine key points on the graph

$$(1, 2), (-1, 2), (2, 5), (-2, 5)$$

Step3: State difference quotient formula

$$DQ = \frac{f(x+h) - f(x)}{h}$$

Step4: Substitute function into the formula

$$DQ = \frac{[3(x+h)^2 - 4] - [3x^2 - 4]}{h}$$

Step5: Expand the numerator expression

$$DQ = \frac{3(x^2 + 2xh + h^2) - 4 - 3x^2 + 4}{h}$$

Step6: Simplify the numerator terms

$$DQ = \frac{6xh + 3h^2}{h}$$

Step7: Divide by h to get final answer

$$DQ = 6x + 3h$$

Answer:

1. The graph of \(g(x) = 3|x| - 1\) is a V-shaped curve with its vertex at \((0, -1)\), passing through the points \((1, 2)\), \((-1, 2)\), \((2, 5)\), and \((-2, 5)\).
2. \(DQ = 6x + 3h\)