Question

What's the common ratio of the geometric sequence 2, 0.5, 0.125, 1/32? Write this geometric sequence as an explicit formula.

a) Common ratio: 1/4, Explicit formula: a_n = 2(1/4)^n
b) Common ratio: 2, Explicit formula: a_n = 2 * 2^n
c) Common ratio: 1/2, Explicit formula: a_n = 2(1/2)^n
d) Common ratio: 1/8, Explicit formula: a_n = 2(1/8)^n

Vertically dilate the linear function z(x) = 4x - 1 by a factor of three. What's the equation of the vertically dilated function?
a) y(x) = 4x - 3
b) y(x) = 12x - 1
c) y(x) = 4x - 1/3
d) y(x) = 4x - 3

Consider the linear functions y(x) = 2x + 8 and g(x) = -2x + 4. What are their y-intercepts? Write these functions in factored form. What are their x-intercepts?
a) y(x) y-intercept: 8, g(x) y-intercept: 4, x-intercepts: None for both
b) y(x) y-intercept: 8, g(x) y-intercept: -4, x-intercepts: None for both
c) y(x) y-intercept: -8, g(x) y-intercept: -4, x-intercepts: None for both
d) y(x) y-intercept: -8, g(x) y-intercept: 4, x-intercepts: None for both

Answer 1

The common ratio of the geometric sequence is 1/4, with an explicit formula of a_n = 2(1/4)^n. The vertically dilated linear function is y(x) = 12x - 3. The y-intercepts for y(x) and g(x) are 8 and 4, respectively, and their factored forms are y(x) = 2(x + 4) and g(x) = -2(x - 2).

Step-by-step explanation:

Geometric Sequence and Common Ratio

The common ratio of a geometric sequence can be determined by dividing any term by the preceding term. In the sequence 2, 0.5, 0.125, 1/32, each term is ⅔ of the previous term, so the common ratio is ⅔. Therefore, the explicit formula for the nth term (a_n) of the sequence would be a_n = 2(⅔)^n. This corresponds with option a).

Vertical Dilation of a Linear Function

To vertically dilate z(x) = 4x - 1 by a factor of three, we multiply the output by 3, leading to the new function y(x) = 12x - 3. So, answer d) is correct.

Y-Intercepts and Factored Form of Linear Functions

The y(x) function has a y-intercept at 8 and an x-intercept when y(x) = 0, which is at x = -4. Thus, in factored form, y(x) can be expressed as y(x) = 2(x + 4). On the other hand, g(x) = -2x + 4 has a y-intercept at 4 and an x-intercept at x = 2. In factored form, g(x) = -2(x - 2). Therefore, the correct option is b).