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For the linear function $f(x)=2-\frac{1}{6} x$ ; (a) evaluate $f(-6)$ and $f(12)$ ; (b) find the zero of $f$ ; and $(c)$ graph $f$ . How can the graph of $f$ be used to determine the zero of $f$ ? (a) $f(-6)=$ $\square$ (Type an integer or a simplified fraction.) $f(12)=$ $\square$ (Type an integer or a simplified fraction.) (b) The zero of $f$ is $\square$ . (Type an integer or a simplified fraction.) (c) Choose the correct graph below. A. B. c. D.
How can the graph of $f$ be used to determine the zero of $f$ ? A. Determine the zero of $f$ by finding the $y$ -coordinate of the point where the line intersects the $y$ -axis. B. Determine the zero of $f$ by finding the $x$ -coordinate of the point where the line intersects the $y$ -axis. C. Determine the zero of $f$ by finding the $y$ -coordinate of the point where the line intersects the $x$ -axis. D. Determine the zero of $f$ by finding the $x$ -coordinate of the point where the line intersects the $x$ -axis.

Studdy Solution

STEP 1

1. We are given a linear function $f(x) = 2 - \frac{1}{6}x$ .
2. We need to evaluate the function at specific points, find the zero of the function, and understand how the graph can help determine the zero.

STEP 2

1. Evaluate the function at given points.
2. Find the zero of the function.
3. Discuss how the graph can be used to determine the zero.

STEP 3

Evaluate the function at $x = -6$ .
$f(-6) = 2 - \frac{1}{6}(-6)$
Calculate $\frac{1}{6}(-6)$ :
$\frac{1}{6}(-6) = -1$
Substitute back into the function:
$f(-6) = 2 - (-1) = 2 + 1 = 3$
So, $f(-6) = 3$ .

STEP 4

Evaluate the function at $x = 12$ .
$f(12) = 2 - \frac{1}{6}(12)$
Calculate $\frac{1}{6}(12)$ :
$\frac{1}{6}(12) = 2$
Substitute back into the function:
$f(12) = 2 - 2 = 0$
So, $f(12) = 0$ .

STEP 5

Find the zero of the function. The zero of the function is the value of $x$ for which $f(x) = 0$ .
Set the function equal to zero:
$0 = 2 - \frac{1}{6}x$
Solve for $x$ :
$\frac{1}{6}x = 2$
Multiply both sides by 6 to isolate $x$ :
$x = 12$
The zero of $f$ is $x = 12$ .

STEP 6

Discuss how the graph can be used to determine the zero.
The zero of the function is the $x$ -coordinate where the graph of the function intersects the $x$ -axis. Therefore, the correct answer is:
D. Determine the zero of $f$ by finding the $x$ -coordinate of the point where the line intersects the $x$ -axis.

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