To describe the graph of the function [tex]\( y = 4(x + 2) \)[/tex] on the interval [tex]\([0, 3)\)[/tex], let's go through a detailed step-by-step explanation:
### 1. Understanding the Function
The function given is [tex]\( y = 4(x + 2) \)[/tex]. This is a linear function since it can be written in the form [tex]\( y = 4x + 8 \)[/tex], which is a straight line where the slope is 4 and the y-intercept is 8.
### 2. Interval Specification
The interval specified is [tex]\([0, 3)\)[/tex]. This means we need to examine the behavior of the function for [tex]\( x \)[/tex] values starting from 0 up to, but not including, 3.
### 3. Calculating Key Points
We can calculate the values of [tex]\( y \)[/tex] at specific [tex]\( x \)[/tex] values within this interval.
- At [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 4(0 + 2) = 4 \cdot 2 = 8
\][/tex]
- At [tex]\( x = 0.1 \)[/tex]:
[tex]\[
y = 4(0.1 + 2) = 4 \cdot 2.1 = 8.4
\][/tex]
- At [tex]\( x = 0.2 \)[/tex]:
[tex]\[
y = 4(0.2 + 2) = 4 \cdot 2.2 = 8.8
\][/tex]
- Continue calculating for other values up to just before [tex]\( x = 3 \)[/tex]:
Here is a selection of calculated points:
[tex]\[
\begin{array}{c|c}
x & y \\
\hline
0.3 & 9.2 \\
0.4 & 9.6 \\
0.5 & 10.0 \\
0.6 & 10.4 \\
0.7 & 10.8 \\
0.8 & 11.2 \\
0.9 & 11.6 \\
1.0 & 12.0 \\
1.1 & 12.4 \\
1.2 & 12.8 \\
1.3 & 13.2 \\
1.4 & 13.6 \\
1.5 & 14.0 \\
1.6 & 14.4 \\
1.7 & 14.8 \\
1.8 & 15.2 \\
1.9 & 15.6 \\
2.0 & 16.0 \\
2.1 & 16.4 \\
2.2 & 16.8 \\
2.3 & 17.2 \\
2.4 & 17.6 \\
2.5 & 18.0 \\
2.6 & 18.4 \\
2.7 & 18.8 \\
2.8 & 19.2 \\
2.9 & 19.6 \\
\end{array}
\][/tex]
### 4. Describing the Graph
- Starting Point: The graph starts at the point (0, 8).
- Behavior: As [tex]\( x \)[/tex] increases from 0 to just below 3, [tex]\( y \)[/tex] increases linearly.
- Slope: The slope of the line is 4, which means for every unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 4 units.
- Linearity: The function is a straight line.
- End Point Threshold: The function values approach [tex]\( y = 4(3 + 2) = 4 \cdot 5 = 20 \)[/tex] as [tex]\( x \)[/tex] approaches 3, but does not include this point, since the interval is open at [tex]\( x = 3 \)[/tex].
### 5. Graph Summary
- Graph Type: Straight Line
- Domain: [tex]\([0, 3)\)[/tex]
- Range within the interval: Starting from [tex]\( y = 8 \)[/tex] at [tex]\( x = 0 \)[/tex] and extending up to just below [tex]\( y = 20 \)[/tex] as [tex]\( x \)[/tex] approaches 3.
- Slope and Trend: The slope is positive, indicating an increasing trend.
In conclusion, the graph of [tex]\( y = 4(x + 2) \)[/tex] over the interval [tex]\([0, 3)\)[/tex] is a straight line starting at (0, 8) and increasing linearly with a constant slope of 4, reaching toward but not touching the point [tex]\((3, 20)\)[/tex].
