Math / Algebra

QuestionFor the linear function $f(x)=2-\frac{1}{3} x$ ; (a) evaluate $f(-3)$ and $f(6)$ ; (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) $f(-3)=$ $\square$ (Type an integer or a simplified fraction.) $f(6)=$ $\square$ (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 $x$ -coordinate of the point where the line intersects the $x$ -axis. D. Determine the zero of $f$ by finding the $y$ -coordinate of the point where the line intersects the $x$ -axis.

Studdy Solution

STEP 1

1. We are given the linear function $f(x) = 2 - \frac{1}{3}x$ .
2. We need to evaluate $f(-3)$ and $f(6)$ .
3. We need to find the zero of the function $f$ .
4. We need to graph the function $f$ .
5. We need to determine how the graph can be used to find the zero of $f$ .

STEP 2

1. Evaluate $f(-3)$ .
2. Evaluate $f(6)$ .
3. Find the zero of the function $f$ .
4. Graph the function $f$ .
5. Determine how the graph can be used to find the zero of $f$ .

STEP 3

Substitute $x = -3$ into the function $f(x)$ :
$f(-3) = 2 - \frac{1}{3}(-3)$

STEP 4

Simplify the expression:
$f(-3) = 2 + 1$ $f(-3) = 3$

STEP 5

Substitute $x = 6$ into the function $f(x)$ :
$f(6) = 2 - \frac{1}{3}(6)$

STEP 6

Simplify the expression:
$f(6) = 2 - 2$ $f(6) = 0$

STEP 7

To find the zero of the function, set $f(x) = 0$ :
$0 = 2 - \frac{1}{3}x$

STEP 8

Solve for $x$ :
$\frac{1}{3}x = 2$ $x = 6$
The zero of $f$ is $x = 6$ .

STEP 9

Graph the function $f(x) = 2 - \frac{1}{3}x$ .
- The y-intercept is at $(0, 2)$ . - The slope is $-\frac{1}{3}$ , which means for every 3 units increase in $x$ , $y$ decreases by 1 unit. - The zero of the function is at $x = 6$ .

STEP 10

Determine how the graph can be used to find the zero of $f$ :
The zero of $f$ can be determined by finding the $x$ -coordinate of the point where the line intersects the $x$ -axis.
The correct choice is: C. Determine the zero of $f$ by finding the $x$ -coordinate of the point where the line intersects the $x$ -axis.
The solutions are: (a) $f(-3) = 3$ , $f(6) = 0$ (b) The zero of $f$ is $6$ . (c) The graph is determined by the slope and intercepts, and choice C explains how to find the zero.

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Studdy Solution

STEP 1

1. We are given the linear function $f(x) = 2 - \frac{1}{3}x$ .
2. We need to evaluate $f(-3)$ and $f(6)$ .
3. We need to find the zero of the function $f$ .
4. We need to graph the function $f$ .
5. We need to determine how the graph can be used to find the zero of $f$ .

STEP 2

1. Evaluate $f(-3)$ .
2. Evaluate $f(6)$ .
3. Find the zero of the function $f$ .
4. Graph the function $f$ .
5. Determine how the graph can be used to find the zero of $f$ .

STEP 3

Substitute $x = -3$ into the function $f(x)$ :
$f(-3) = 2 - \frac{1}{3}(-3)$

STEP 4

Simplify the expression:
$f(-3) = 2 + 1$ $f(-3) = 3$

STEP 5

Substitute $x = 6$ into the function $f(x)$ :
$f(6) = 2 - \frac{1}{3}(6)$

STEP 6

Simplify the expression:
$f(6) = 2 - 2$ $f(6) = 0$

STEP 7

To find the zero of the function, set $f(x) = 0$ :
$0 = 2 - \frac{1}{3}x$

STEP 8

Solve for $x$ :
$\frac{1}{3}x = 2$ $x = 6$
The zero of $f$ is $x = 6$ .

STEP 9

Graph the function $f(x) = 2 - \frac{1}{3}x$ .
- The y-intercept is at $(0, 2)$ . - The slope is $-\frac{1}{3}$ , which means for every 3 units increase in $x$ , $y$ decreases by 1 unit. - The zero of the function is at $x = 6$ .

STEP 10

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Studdy Solution

STEP 1

STEP 2

1. Evaluate f(−3) f(-3) f(−3).2. Evaluate f(6) f(6) f(6).3. Find the zero of the function f f f.4. Graph the function f f f.5. Determine how the graph can be used to find the zero of f f f.

STEP 3

Substitute x=−3 x = -3 x=−3 into the function f(x) f(x) f(x):f(−3)=2−13(−3) f(-3) = 2 - \frac{1}{3}(-3) f(−3)=2−31​(−3)

STEP 4

Simplify the expression:f(−3)=2+1 f(-3) = 2 + 1 f(−3)=2+1 f(−3)=3 f(-3) = 3 f(−3)=3

STEP 5

Substitute x=6 x = 6 x=6 into the function f(x) f(x) f(x):f(6)=2−13(6) f(6) = 2 - \frac{1}{3}(6) f(6)=2−31​(6)

STEP 6

Simplify the expression:f(6)=2−2 f(6) = 2 - 2 f(6)=2−2 f(6)=0 f(6) = 0 f(6)=0

STEP 7

To find the zero of the function, set f(x)=0 f(x) = 0 f(x)=0:0=2−13x 0 = 2 - \frac{1}{3}x 0=2−31​x

STEP 8

Solve for x x x:13x=2 \frac{1}{3}x = 2 31​x=2 x=6 x = 6 x=6The zero of f f f is x=6 x = 6 x=6.

STEP 9

Graph the function f(x)=2−13x f(x) = 2 - \frac{1}{3}x f(x)=2−31​x. - The y-intercept is at (0,2) (0, 2) (0,2). - The slope is −13-\frac{1}{3}−31​, which means for every 3 units increase in x x x, y y y decreases by 1 unit. - The zero of the function is at x=6 x = 6 x=6.

STEP 10

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1. Evaluate $f(-3)$ .
2. Evaluate $f(6)$ .
3. Find the zero of the function $f$ .
4. Graph the function $f$ .
5. Determine how the graph can be used to find the zero of $f$ .

Substitute $x = -3$ into the function $f(x)$ :
$f(-3) = 2 - \frac{1}{3}(-3)$

Simplify the expression:
$f(-3) = 2 + 1$ $f(-3) = 3$

Substitute $x = 6$ into the function $f(x)$ :
$f(6) = 2 - \frac{1}{3}(6)$

Simplify the expression:
$f(6) = 2 - 2$ $f(6) = 0$

To find the zero of the function, set $f(x) = 0$ :
$0 = 2 - \frac{1}{3}x$

Solve for $x$ :
$\frac{1}{3}x = 2$ $x = 6$
The zero of $f$ is $x = 6$ .

Graph the function $f(x) = 2 - \frac{1}{3}x$ .
- The y-intercept is at $(0, 2)$ . - The slope is $-\frac{1}{3}$ , which means for every 3 units increase in $x$ , $y$ decreases by 1 unit. - The zero of the function is at $x = 6$ .