Question
Jack and Nina are graphing two equations on a coordinate grid. Jack has graphed the equation y = 2^x
.
If Nina graphs y = 5^x
, where will her graph be in relation to the graph Jack made?
Responses
A For all x > 0 the graph will be higher.For all x > 0 the graph will be higher.
B For all x > 0 the graph will be lower.For all x > 0 the graph will be lower.
C For all x the graph will be higher.For all x the graph will be higher.
D For all x the graph will be lower.
.
If Nina graphs y = 5^x
, where will her graph be in relation to the graph Jack made?
Responses
A For all x > 0 the graph will be higher.For all x > 0 the graph will be higher.
B For all x > 0 the graph will be lower.For all x > 0 the graph will be lower.
C For all x the graph will be higher.For all x the graph will be higher.
D For all x the graph will be lower.
Answers
Answered by
GPT-4o mini
To compare the graphs of the equations \( y = 2^x \) and \( y = 5^x \), we can analyze their behavior as \( x \) changes.
1. **For \( x < 0 \)**: Both \( y = 2^x \) and \( y = 5^x \) will yield values between 0 and 1 (since both bases are greater than 1). Specifically, since \( 5^x \) approaches 0 more rapidly than \( 2^x \) (because \( 5^x < 2^x \) for negative \( x \)), we can conclude that:
- \( 5^x < 2^x \) for all \( x < 0 \).
2. **For \( x = 0 \)**: Both functions equal 1, since \( y = 2^0 = 1 \) and \( y = 5^0 = 1 \). Thus,
- \( 2^0 = 5^0 \).
3. **For \( x > 0 \)**: As \( x \) increases, \( 5^x \) grows faster than \( 2^x \) because the base of \( 5 \) is greater than that of \( 2 \). This implies:
- \( 5^x > 2^x \) for all \( x > 0 \).
Based on the above analysis, we can summarize the position of Nina's graph relative to Jack's graph:
- For \( x < 0 \): \( 5^x < 2^x \) (Nina's graph is lower)
- For \( x = 0 \): \( 5^0 = 2^0 \) (equal)
- For \( x > 0 \): \( 5^x > 2^x \) (Nina's graph is higher)
Given this information, the correct response is:
**A**: For all \( x > 0 \), the graph will be higher.
1. **For \( x < 0 \)**: Both \( y = 2^x \) and \( y = 5^x \) will yield values between 0 and 1 (since both bases are greater than 1). Specifically, since \( 5^x \) approaches 0 more rapidly than \( 2^x \) (because \( 5^x < 2^x \) for negative \( x \)), we can conclude that:
- \( 5^x < 2^x \) for all \( x < 0 \).
2. **For \( x = 0 \)**: Both functions equal 1, since \( y = 2^0 = 1 \) and \( y = 5^0 = 1 \). Thus,
- \( 2^0 = 5^0 \).
3. **For \( x > 0 \)**: As \( x \) increases, \( 5^x \) grows faster than \( 2^x \) because the base of \( 5 \) is greater than that of \( 2 \). This implies:
- \( 5^x > 2^x \) for all \( x > 0 \).
Based on the above analysis, we can summarize the position of Nina's graph relative to Jack's graph:
- For \( x < 0 \): \( 5^x < 2^x \) (Nina's graph is lower)
- For \( x = 0 \): \( 5^0 = 2^0 \) (equal)
- For \( x > 0 \): \( 5^x > 2^x \) (Nina's graph is higher)
Given this information, the correct response is:
**A**: For all \( x > 0 \), the graph will be higher.