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 $f(x)=0$: $$0=-5x+35x=3x=\frac{3}{5}$$ So the x-intercept is $(\frac{3}{5},0)$.

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).