Question: Grace is comparing cell phone plans. A prepaid phone plan costs $0.20$ per minute and has no monthly fee. A contracted phone plan costs $50$ per month and $0.02$ per minute. How will the graphs of the monthly costs of the two cell phone plans compare if $x$ represents minutes purchased in a month?
1. **State the problem:**
Grace wants to compare two cell phone plans based on their monthly cost depending on the number of minutes $x$ used.
2. **Define variables and formulas:**
Let $x$ be the number of minutes used in a month.
- Prepaid plan cost: $$y_1 = 0.20x$$ (no monthly fee, cost per minute $0.20$)
- Contracted plan cost: $$y_2 = 50 + 0.02x$$ (monthly fee $50$, cost per minute $0.02$)
3. **Analyze the graphs:**
- The prepaid plan graph is a line through the origin with slope $0.20$.
- The contracted plan graph is a line with y-intercept $50$ and slope $0.02$.
4. **Compare slopes (steepness):**
- Slope of prepaid plan: $0.20$
- Slope of contracted plan: $0.02$
Since $0.20 > 0.02$, the prepaid plan line is steeper.
5. **Compare y-intercepts:**
- Prepaid plan y-intercept: $0$
- Contracted plan y-intercept: $50$
The contracted plan has a higher y-intercept.
6. **Conclusion:**
The contracted phone plan will have a less steep line and a higher y-intercept compared to the prepaid plan.
**Answer:** The prepaid phone plan will have a steeper line and the contracted phone plan will have a higher y-intercept.
This matches the option: "The prepaid phone plan will have a steeper line and lower y-intercept."