To solve this problem, we start by defining the height function [tex]\( f(x) \)[/tex] based on the given equation for the height of the plant. The equation provided is:
[tex]\[ h = 0.5d + 4 \][/tex]
Here, [tex]\( h \)[/tex] represents the height of the plant, and [tex]\( d \)[/tex] represents the number of days. We can rewrite this equation where [tex]\( f(x) \)[/tex] is the height function and [tex]\( x \)[/tex] is the time in days:
[tex]\[ f(x) = 0.5x + 4 \][/tex]
Now, we need to use this function rule to complete the table for given values of [tex]\( x \)[/tex]. These values of [tex]\( x \)[/tex] range from 0 to 6. We will substitute each value of [tex]\( x \)[/tex] into the function [tex]\( f(x) = 0.5x + 4 \)[/tex] to find the corresponding height [tex]\( f(x) \)[/tex].
1. When [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 0.5 \cdot 0 + 4 = 0 + 4 = 4.0 \][/tex]
2. When [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 0.5 \cdot 1 + 4 = 0.5 + 4 = 4.5 \][/tex]
3. When [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = 0.5 \cdot 2 + 4 = 1 + 4 = 5.0 \][/tex]
4. When [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = 0.5 \cdot 3 + 4 = 1.5 + 4 = 5.5 \][/tex]
5. When [tex]\( x = 4 \)[/tex]:
[tex]\[ f(4) = 0.5 \cdot 4 + 4 = 2 + 4 = 6.0 \][/tex]
6. When [tex]\( x = 5 \)[/tex]:
[tex]\[ f(5) = 0.5 \cdot 5 + 4 = 2.5 + 4 = 6.5 \][/tex]
7. When [tex]\( x = 6 \)[/tex]:
[tex]\[ f(6) = 0.5 \cdot 6 + 4 = 3 + 4 = 7.0 \][/tex]
Let's fill in the table with these calculated values:
[tex]\[
\begin{array}{|l|l|l|l|l|l|l|l|}
\hline
x & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
f(x) & 4.0 & 4.5 & 5.0 & 5.5 & 6.0 & 6.5 & 7.0 \\
\hline
\end{array}
\][/tex]
Next, we would use the drawing tools to plot these points and create a graph representing this relationship. The points [tex]\((x, f(x))\)[/tex] that we will plot are:
[tex]\[ (0, 4.0), (1, 4.5), (2, 5.0), (3, 5.5), (4, 6.0), (5, 6.5), (6, 7.0) \][/tex]
To draw the graph:
1. Plot each point on a coordinate plane.
2. Connect the points with a straight line since this is a linear function.
The graph will show a straight line rising from the point [tex]\((0, 4)\)[/tex] up to the point [tex]\((6, 7)\)[/tex], depicting the linear relationship between the number of days and the height of the plant.
