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To graph the function [tex]\( f(x) = -\frac{1}{3}x + 8 \)[/tex], follow these steps:
### 1. Determine Key Points on the Graph
First, identify the y-intercept and a few other points to draw the graph.
- Y-intercept: The y-intercept occurs where [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = -\frac{1}{3}(0) + 8 = 8 \][/tex]
So, the y-intercept is [tex]\( (0, 8) \)[/tex].
- X-intercept: This is where [tex]\( f(x) = 0 \)[/tex].
[tex]\[ 0 = -\frac{1}{3}x + 8 \implies \frac{1}{3}x = 8 \implies x = 24 \][/tex]
So, the x-intercept is [tex]\( (24, 0) \)[/tex].
- Additional Points: Choose values for [tex]\( x \)[/tex] to calculate corresponding [tex]\( y \)[/tex] values.
For example:
[tex]\[ \begin{aligned} &f(-6) = -\frac{1}{3}(-6) + 8 = 2 + 8 = 10\\ &f(6) = -\frac{1}{3}(6) + 8 = -2 + 8 = 6 \end{aligned} \][/tex]
By plugging in, we get additional points such as [tex]\( (-6, 10) \)[/tex] and [tex]\( (6, 6) \)[/tex].
### 2. Plot the Points on the Graph
Plot the points found on a coordinate grid:
- [tex]\( (0, 8) \)[/tex] (the y-intercept)
- [tex]\( (24, 0) \)[/tex] (the x-intercept)
- [tex]\( (-6, 10) \)[/tex]
- [tex]\( (6, 6) \)[/tex]
### 3. Draw the Line
Draw a straight line through these points. This line represents the graph of the function [tex]\( f(x) = -\frac{1}{3}x + 8 \)[/tex].
### Graph Preview
1. Plot the y-intercept (0, 8)
2. Plot the x-intercept (24, 0)
3. Plot additional points (-6, 10) and (6, 6)
4. Draw the line through these points
Here is a sketch of the graph:
[tex]\[ \begin{array}{cc} \begin{tikzpicture}[scale=0.8] \draw[<->,thick] (-10,0)--(30,0) node[anchor=north]{$x$}; \draw[<->,thick] (0,-3)--(0,12) node[anchor=east]{$f(x)$}; % Grid \draw[lightgray,thin] (-10,-3) grid (30,12); % Points \filldraw[blue] (0,8) circle (3pt) node[anchor=east] {$(0,8)$}; \filldraw[red] (24,0) circle (3pt) node[anchor=north] {$(24,0)$}; \filldraw[green] (-6,10) circle (3pt) node[anchor=east] {$(-6, 10)$}; \filldraw[orange] (6,6) circle (3pt) node[anchor=north] {$(6,6)$}; % Line f(x) = -1/3x + 8 \draw[very thick,cyan] (-10,11.33) -- (30, -2); % Labels \node at (15,8) {f(x) = $-\frac{1}{3}x + 8$}; \end{tikzpicture} \end{array} \][/tex]
This line accurately represents the function [tex]\( f(x) = -\frac{1}{3}x + 8 \)[/tex].
### 1. Determine Key Points on the Graph
First, identify the y-intercept and a few other points to draw the graph.
- Y-intercept: The y-intercept occurs where [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = -\frac{1}{3}(0) + 8 = 8 \][/tex]
So, the y-intercept is [tex]\( (0, 8) \)[/tex].
- X-intercept: This is where [tex]\( f(x) = 0 \)[/tex].
[tex]\[ 0 = -\frac{1}{3}x + 8 \implies \frac{1}{3}x = 8 \implies x = 24 \][/tex]
So, the x-intercept is [tex]\( (24, 0) \)[/tex].
- Additional Points: Choose values for [tex]\( x \)[/tex] to calculate corresponding [tex]\( y \)[/tex] values.
For example:
[tex]\[ \begin{aligned} &f(-6) = -\frac{1}{3}(-6) + 8 = 2 + 8 = 10\\ &f(6) = -\frac{1}{3}(6) + 8 = -2 + 8 = 6 \end{aligned} \][/tex]
By plugging in, we get additional points such as [tex]\( (-6, 10) \)[/tex] and [tex]\( (6, 6) \)[/tex].
### 2. Plot the Points on the Graph
Plot the points found on a coordinate grid:
- [tex]\( (0, 8) \)[/tex] (the y-intercept)
- [tex]\( (24, 0) \)[/tex] (the x-intercept)
- [tex]\( (-6, 10) \)[/tex]
- [tex]\( (6, 6) \)[/tex]
### 3. Draw the Line
Draw a straight line through these points. This line represents the graph of the function [tex]\( f(x) = -\frac{1}{3}x + 8 \)[/tex].
### Graph Preview
1. Plot the y-intercept (0, 8)
2. Plot the x-intercept (24, 0)
3. Plot additional points (-6, 10) and (6, 6)
4. Draw the line through these points
Here is a sketch of the graph:
[tex]\[ \begin{array}{cc} \begin{tikzpicture}[scale=0.8] \draw[<->,thick] (-10,0)--(30,0) node[anchor=north]{$x$}; \draw[<->,thick] (0,-3)--(0,12) node[anchor=east]{$f(x)$}; % Grid \draw[lightgray,thin] (-10,-3) grid (30,12); % Points \filldraw[blue] (0,8) circle (3pt) node[anchor=east] {$(0,8)$}; \filldraw[red] (24,0) circle (3pt) node[anchor=north] {$(24,0)$}; \filldraw[green] (-6,10) circle (3pt) node[anchor=east] {$(-6, 10)$}; \filldraw[orange] (6,6) circle (3pt) node[anchor=north] {$(6,6)$}; % Line f(x) = -1/3x + 8 \draw[very thick,cyan] (-10,11.33) -- (30, -2); % Labels \node at (15,8) {f(x) = $-\frac{1}{3}x + 8$}; \end{tikzpicture} \end{array} \][/tex]
This line accurately represents the function [tex]\( f(x) = -\frac{1}{3}x + 8 \)[/tex].
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