Question

questions:

1. both shifts earn $30 after 5 hours of work, but the graphs look different. how can you tell which shift represents a

proportional relationship, and which one does not? what does that mean in this situation?
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1. write the equation for each shift. then, describe the slope and y - intercept for the after school shift. what does each

part of the equation tell you about milos earnings?
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milo runs a small cookie stand at his school. he sells cookies during lunch and after school. his friend made a graph to help
him keep track of how much money he makes from each shift.
cookie stand earnings
questions:

1. both shifts earn $30 after 5 hours of work, but the graphs look different. how can you tell which shift represents a

proportional relationship, and which one does not? what does that mean in this situation?
chat with a tutor

Explanation:

Step1: Determine proportional relationship

A proportional relationship graph passes through the origin \((0,0)\). The Lunch Shift graph passes through \((0,0)\), so it is proportional. The After - School Shift graph does not pass through \((0,0)\), so it is non - proportional. In this situation, for the Lunch Shift (proportional), Milo starts earning money immediately (\(y = 0\) when \(x = 0\), no initial amount). For the After - School Shift (non - proportional), there is an initial amount of money (y - intercept) before he starts working (even at \(x = 0\), \(y>0\)).

Step2: Find equations

For the Lunch Shift (proportional, \(y=kx\)). Using the point \((5,30)\), substitute into \(y = kx\): \(30=k\times5\), so \(k = 6\). The equation is \(y = 6x\). The slope \(k = 6\) means Milo earns \( \$6\) per hour during the Lunch Shift.
For the After - School Shift (linear, \(y=mx + b\)). Let's assume two points: when \(x = 0\), \(y=b\) (y - intercept). When \(x = 5\), \(y = 30\). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's assume another point \((0,b)\) and \((5,30)\). If we assume the slope \(m\) (using the fact that for the same \(x = 5\) output \(y = 30\) as in the Lunch Shift, but with a non - zero intercept). Let's use the formula \(y=mx + b\). We know that when \(x = 5\), \(y = 30\). If we assume the slope \(m\) (using the rate of change). Let's say the After - School Shift: assume \(y=mx + b\). We know that when \(x = 0\), \(y=b\) (initial amount). When \(x = 5\), \(y=30\). The slope \(m=\frac{30 - b}{5}\). If we assume from the graph (visually, if we consider the non - zero intercept). Let's assume \(b = 6\) (for example, if we assume a point \((0,6)\) and \((5,30)\)), then \(m=\frac{30 - 6}{5}=\frac{24}{5}=4.8\). The equation is \(y = 4.8x+6\). The slope \(m = 4.8\) means Milo earns \( \$4.8\) per hour during the After - School Shift, and the \(y\) - intercept \(b = 6\) means he has an initial \( \$6\) (maybe from a previous day's savings or a starting fund) before he starts working the After - School Shift.

Answer:

1. The Lunch Shift represents a proportional relationship (passes through \((0,0)\)), the After - School Shift does not (does not pass through \((0,0)\)). For the Lunch Shift, Milo has no initial earnings (starts from \(0\) when \(0\) hours worked). For the After - School Shift, there is an initial amount of money before he starts working.
2. Lunch Shift equation: \(y = 6x\) (slope \(=6\), means \( \$6\) per hour, \(y\) - intercept \(=0\), no initial amount). After - School Shift equation (example assuming \(y\) - intercept \(=6\)): \(y = 4.8x + 6\) (slope \(=4.8\), means \( \$4.8\) per hour, \(y\) - intercept \(=6\), initial amount of \( \$6\)).