Question

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