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

Answer 1

To analyze the linear function \( f(x) = \frac{1}{3} (3x + 9) - 6x \), we can simplify it first.

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

2. Combine like terms:
\[
f(x) = -5x + 3
\]

Now that we have the function in the slope-intercept form \( f(x) = -5x + 3 \), we can find the key features:

### Finding the y-intercept
The y-intercept occurs when \( x = 0 \):
\[
f(0) = -5(0) + 3 = 3
\]
So the y-intercept is \( (0, 3) \).

### Finding the x-intercept
The x-intercept occurs when \( f(x) = 0 \):
\[
0 = -5x + 3
\]
Solving for \( x \):
\[
5x = 3 \quad \Rightarrow \quad x = \frac{3}{5}
\]
So the x-intercept is \( \left(\frac{3}{5}, 0\right) \).

### Evaluation of the provided choices:
- **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**

Therefore, the true key feature of the graph is:
**B: y-intercept (0, 3)**.