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Jessica is selling books during the summer to earn money for college. She earns a commission on each sale but has to pay for her own expenses.

After a month of driving from neighborhood to neighborhood and walking door-to-door, she figures out that her weekly earnings are approximately a linear function of the number of doors she knocks on.

She writes the equation of the function like this: [tex]$E(x)=20x-50$[/tex], where [tex]$x$[/tex] is the number of doors she knocks on during the week and [tex][tex]$E(x)$[/tex][/tex] is her earnings for the week in dollars.

Which of the following are reasonable interpretations of the [tex]$y$[/tex]-intercept of Jessica's function?

Check all that apply.
A. Her expenses are [tex][tex]$\$[/tex]50$[/tex] per week.
B. If she does not knock on any doors at all during the week, she will lose [tex]$\[tex]$50$[/tex][/tex].
C. She can earn [tex]$\$20$[/tex] per week even if she does not knock on any doors.
D. She will lose [tex]$\[tex]$20$[/tex][/tex] per week if she does not knock on any doors.



Answer :

GinnyAnswer
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.

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