Sketch the linear function f(x) = 1/3 (3x + 9) − 6x. Select the TRUE key feature of the graph.

Question

Responses
A x-intercept (6, 0
B y-intercept (0, 3)
C y-intercept (0, 1/3)
D x-intercept (1/3, 0)

Answer 1

To analyze the function \( f(x) = \frac{1}{3}(3x + 9) - 6x \) and find its key features, we first simplify it.

1. Distribute the \(\frac{1}{3}\): \[ f(x) = \frac{1}{3} \cdot 3x + \frac{1}{3} \cdot 9 - 6x = x + 3 - 6x = -5x + 3 \]

Now we have a linear function: \[ f(x) = -5x + 3 \]

Finding the y-intercept:

To find the y-intercept, set \( x = 0 \): \[ f(0) = -5(0) + 3 = 3 \] Thus, the \( y \)-intercept is \( (0, 3) \).

Finding the x-intercept:

To find the x-intercept, set \( f(x) = 0 \): \[ 0 = -5x + 3 \ 5x = 3 \ x = \frac{3}{5} \] So the x-intercept is \( \left( \frac{3}{5}, 0 \right) \).

Examining the options:

• A. x-intercept (6, 0): False
• B. y-intercept (0, 3): True
• C. y-intercept (0, 1/3): False
• D. x-intercept (1/3, 0): False

The TRUE key feature of the graph is: B. y-intercept (0, 3).