Answer :
Let's analyze the given function [tex]\( E(x) = 20x - 50 \)[/tex] to understand the interpretations of the [tex]\( y \)[/tex]-intercept.
The function [tex]\( E(x) \)[/tex] is a linear equation where:
- [tex]\( x \)[/tex] represents the number of doors Jessica knocks on in a week.
- [tex]\( E(x) \)[/tex] represents her weekly earnings in dollars.
The [tex]\( y \)[/tex]-intercept of a linear function [tex]\( E(x) = mx + b \)[/tex] is the value when [tex]\( x = 0 \)[/tex]. Here, [tex]\( b \)[/tex] is the [tex]\( y \)[/tex]-intercept.
1. Identify the [tex]\( y \)[/tex]-intercept:
- In the given function, [tex]\( E(x) = 20x - 50 \)[/tex], the [tex]\( y \)[/tex]-intercept is [tex]\(-50\)[/tex].
So, when Jessica knocks on 0 doors ([tex]\( x = 0 \)[/tex]):
[tex]\[ E(0) = 20 \cdot 0 - 50 = -50 \][/tex]
2. Interpret the [tex]\( y \)[/tex]-intercept:
- When [tex]\( x = 0 \)[/tex], [tex]\( E(x) = -50 \)[/tex]. This means that if Jessica does not knock on any doors during the week, her earnings ([tex]\( E \)[/tex]) will be [tex]\(-\$50\)[/tex].
Now we evaluate the presented options:
- Option A: Her expenses are \[tex]$50 per week. This can be seen as valid because the intercept suggests she starts with a -\$[/tex]50 deficit. This might imply that her weekly cost or expense is \[tex]$50. - Option B: If she does not knock on any doors during the week, she will lose \$[/tex]50.
This is also valid. If Jessica does not knock on any doors, [tex]\( E(0) = -50 \)[/tex] indicates she will be [tex]\(-\$50\)[/tex], meaning a \[tex]$50 loss. - Option C: She can earn \$[/tex]20 per week even if she does not knock on any doors.
This is incorrect. The function shows a negative earning when [tex]\( x = 0 \)[/tex], so she does not earn anything by default without knocking on doors.
- Option D: She will lose \[tex]$20 per week if she does not knock on any doors. This is incorrect. The loss is \( \$[/tex]50 \), not \[tex]$20. Based on the function and its interpretation: The reasonable interpretations of the \( y \)-intercept of Jessica's function are: - A. Her expenses are \$[/tex]50 per week.
- B. If she does not knock on any doors at all during the week, she will lose \$50.
Therefore, the correct interpretations are options A and B.
The function [tex]\( E(x) \)[/tex] is a linear equation where:
- [tex]\( x \)[/tex] represents the number of doors Jessica knocks on in a week.
- [tex]\( E(x) \)[/tex] represents her weekly earnings in dollars.
The [tex]\( y \)[/tex]-intercept of a linear function [tex]\( E(x) = mx + b \)[/tex] is the value when [tex]\( x = 0 \)[/tex]. Here, [tex]\( b \)[/tex] is the [tex]\( y \)[/tex]-intercept.
1. Identify the [tex]\( y \)[/tex]-intercept:
- In the given function, [tex]\( E(x) = 20x - 50 \)[/tex], the [tex]\( y \)[/tex]-intercept is [tex]\(-50\)[/tex].
So, when Jessica knocks on 0 doors ([tex]\( x = 0 \)[/tex]):
[tex]\[ E(0) = 20 \cdot 0 - 50 = -50 \][/tex]
2. Interpret the [tex]\( y \)[/tex]-intercept:
- When [tex]\( x = 0 \)[/tex], [tex]\( E(x) = -50 \)[/tex]. This means that if Jessica does not knock on any doors during the week, her earnings ([tex]\( E \)[/tex]) will be [tex]\(-\$50\)[/tex].
Now we evaluate the presented options:
- Option A: Her expenses are \[tex]$50 per week. This can be seen as valid because the intercept suggests she starts with a -\$[/tex]50 deficit. This might imply that her weekly cost or expense is \[tex]$50. - Option B: If she does not knock on any doors during the week, she will lose \$[/tex]50.
This is also valid. If Jessica does not knock on any doors, [tex]\( E(0) = -50 \)[/tex] indicates she will be [tex]\(-\$50\)[/tex], meaning a \[tex]$50 loss. - Option C: She can earn \$[/tex]20 per week even if she does not knock on any doors.
This is incorrect. The function shows a negative earning when [tex]\( x = 0 \)[/tex], so she does not earn anything by default without knocking on doors.
- Option D: She will lose \[tex]$20 per week if she does not knock on any doors. This is incorrect. The loss is \( \$[/tex]50 \), not \[tex]$20. Based on the function and its interpretation: The reasonable interpretations of the \( y \)-intercept of Jessica's function are: - A. Her expenses are \$[/tex]50 per week.
- B. If she does not knock on any doors at all during the week, she will lose \$50.
Therefore, the correct interpretations are options A and B.
