Sure! Let's go through the steps to graph the given cost function [tex]\(d(x) = 3x + 2.00\)[/tex] and determine the proper graph and corresponding axis labels.
### Step-by-Step Solution:
1. Understand the Equation:
[tex]\[
d(x) = 3x + 2.00
\][/tex]
Here, [tex]\(d(x)\)[/tex] represents the cost of the taxi ride in dollars, and [tex]\(x\)[/tex] represents the number of minutes.
2. Identify Key Components of the Equation:
- The equation is in slope-intercept form [tex]\(y = mx + b\)[/tex], where:
- [tex]\(m = 3\)[/tex] (slope)
- [tex]\(b = 2.00\)[/tex] (y-intercept)
3. Y-Intercept:
- At [tex]\(x = 0\)[/tex], calculate [tex]\(d(x)\)[/tex]:
[tex]\[
d(0) = 3(0) + 2.00 = 2.00
\][/tex]
- The y-intercept is [tex]\((0, 2.00)\)[/tex]. This means that when the taxi ride duration is 0 minutes, the cost is [tex]$2.00 (which is likely the base fare).
4. Slope:
- The slope \(m = 3\) means that for every additional minute (\(x\)), the cost (\(d(x)\)) increases by \$[/tex]3.00.
5. Plot Key Points:
- First Point (Y-Intercept): [tex]\((0, 2.00)\)[/tex]
- Second Point: Choose a value for [tex]\(x\)[/tex]. For instance, let [tex]\(x = 1\)[/tex]:
[tex]\[
d(1) = 3(1) + 2.00 = 5.00
\][/tex]
So, the point is [tex]\((1, 5.00)\)[/tex].
- Third Point: Choose another value for [tex]\(x\)[/tex]. Let [tex]\(x = 2\)[/tex]:
[tex]\[
d(2) = 3(2) + 2.00 = 8.00
\][/tex]
So, the point is [tex]\((2, 8.00)\)[/tex].
6. Draw the Graph:
- Mark the y-axis starting from 2.00 (base fare).
- Plot the points [tex]\((0, 2.00)\)[/tex], [tex]\((1, 5.00)\)[/tex], and [tex]\((2, 8.00)\)[/tex].
- Draw a straight line through these points.
7. Label the Axes:
- The x-axis represents the number of minutes ([tex]\(x\)[/tex]).
- The y-axis represents the cost in dollars ([tex]\(d(x)\)[/tex]).
Based on the graph and labels above, the correct graph will have the following features:
- A straight line passing through the points [tex]\((0, 2.00)\)[/tex], [tex]\((1, 5.00)\)[/tex], and [tex]\((2, 8.00)\)[/tex].
- The x-axis labeled as "Minutes".
- The y-axis labeled as "Cost in Dollars".
### Conclusion:
After plotting and analyzing the graph of [tex]\(d(x) = 3x + 2.00\)[/tex], we have identified the key features and labeled the axes correctly. Based on these characteristics, look for a graph that correctly depicts this behavior with the appropriate axis labels.
