svlc algebra 1a - standard (15260)
special linear relationships
which graphs show functions with direct variation? choose three correct answers.

graph 1: parking garage rates, points (1,2.4), (5,4)
graph 2: ferry ride cost, points (1,2), (5,8)
graph 3: breakfast cost, points (1,2), (6,6)
graph 4: cost of cinnamon, points (1,0.3), (5,1.5)
graph 5: euro to u.s dollar conversion, points (1,1.4), (5,7)

Question

svlc algebra 1a - standard (15260)
special linear relationships
which graphs show functions with direct variation? choose three correct answers.

graph 1: parking garage rates, points (1,2.4), (5,4)
graph 2: ferry ride cost, points (1,2), (5,8)
graph 3: breakfast cost, points (1,2), (6,6)
graph 4: cost of cinnamon, points (1,0.3), (5,1.5)
graph 5: euro to u.s dollar conversion, points (1,1.4), (5,7)

Accepted Answer

# Brief Explanations:
A direct variation function has the form $ y = kx $, meaning the graph is a straight line passing through the origin $(0,0)$ (since when $ x = 0 $, $ y = 0 $) and the ratio $ \frac{y}{x} $ is constant (the constant of variation $ k $).

1. **Ferry Ride Cost**: The line passes through $(1,2)$ and $(5,8)$. Check the ratio: $ \frac{2}{1}=2 $ and $ \frac{8}{5} = 1.6 $? Wait, no—wait, actually, let's re - check. Wait, the graph for Ferry Ride Cost: when $ x = 1 $, $ y = 2 $; $ x = 5 $, $ y = 8 $. The slope $ k=\frac{y}{x}$, for $ x = 1 $, $ k = 2 $; for $ x = 5 $, $ k=\frac{8}{5}=1.6 $? Wait, no, maybe I misread. Wait, no, the Breakfast Cost graph: passes through $(0,0)$ (since when $ x = 0 $, $ y = 0 $) and $(1,2)$, $(6,12)$? Wait, the Breakfast Cost graph has a point $(1,2)$ and $(6,12)$, and it passes through the origin. The ratio $ \frac{y}{x}=\frac{2}{1}=2$, $ \frac{12}{6}=2 $, so it's direct variation.

2. **Breakfast Cost**: The line passes through the origin $(0,0)$ (we can see that when $ x = 0 $, $ y = 0 $) and points like $(1,2)$, $(6,12)$. The ratio $ \frac{y}{x}=\frac{2}{1}=2$, $ \frac{12}{6}=2 $, so it's a direct variation ( $ y = 2x $ ).

3. **Euro to U.S Dollar Conversion**: The line passes through $(0,0)$ (since when $ x = 0 $ (0 Euros), $ y = 0 $ (0 US Dollars)) and $(1,1.4)$, $(5,7)$. The ratio $ \frac{y}{x}=\frac{1.4}{1}=1.4$, $ \frac{7}{5}=1.4 $, so it's direct variation ( $ y = 1.4x $ ).

4. **Ferry Ride Cost**: Wait, no, the Ferry Ride Cost graph: when $ x = 0 $, does $ y = 0 $? The graph shows a y - intercept above 0? Wait, no, maybe I made a mistake. Wait, the Parking Garage Rates graph: has a y - intercept of 2 (when $ x = 0 $, $ y = 2 $), so it's not direct variation (since direct variation must pass through the origin). The Cost of Cinnamon graph: when $ x = 0 $, $ y = 0 $? Wait, the Cost of Cinnamon graph has a point $(1,0.3)$ and $(5,1.5)$, and it passes through the origin? Wait, the point $(0,0)$ is on it? Let's check the ratio: $ \frac{0.3}{1}=0.3$, $ \frac{1.5}{5}=0.3 $, so it is direct variation. Wait, I think I messed up earlier.

Wait, let's re - evaluate each graph:

- **Parking Garage Rates**: The line has a y - intercept of 2 (when $ x = 0 $, $ y = 2 $), so it's not direct variation (since $ y=kx + b$ with $ b
eq0$).

- **Ferry Ride Cost**: Let's check the y - intercept. If the line does not pass through the origin, it's not direct variation. Wait, the graph for Ferry Ride Cost: when $ x = 0 $, what's $ y $? The graph shows that when $ x = 0 $, $ y$ is not 0 (it looks like it starts at $ y = 1 $ or something? No, maybe the Breakfast Cost, Cost of Cinnamon, and Euro to U.S Dollar Conversion are the correct ones. Wait, the Breakfast Cost graph: passes through (0,0), (1,2), (6,12) – direct variation. Cost of Cinnamon: passes through (0,0), (1,0.3), (5,1.5) – $ \frac{0.3}{1}=0.3$, $ \frac{1.5}{5}=0.3 $ – direct variation. Euro to U.S Dollar Conversion: passes through (0,0), (1,1.4), (5,7) – $ \frac{1.4}{1}=1.4$, $ \frac{7}{5}=1.4 $ – direct variation. Wait, but the problem says "choose three correct answers". Let's check the ratios:

- Breakfast Cost: $ \frac{y}{x}=\frac{2}{1}=2$, $ \frac{12}{6}=2 $ (assuming the upper point is (6,12)) – direct variation (passes through origin).

- Cost of Cinnamon: $ \frac{y}{x}=\frac{0.3}{1}=0.3$, $ \frac{1.5}{5}=0.3 $ – direct variation (passes through origin).

- Euro to U.S Dollar Conversion: $ \frac{y}{x}=\frac{1.4}{1}=1.4$, $ \frac{7}{5}=1.4 $ – direct variation (passes through origin).

Wait, maybe the Ferry Ride Cost graph does not pass through the origin. Let's check the Parking Garage Rates: y - intercept is 2, so not direct. Ferry Ride Cost: let's see the y - intercept. If the line for Ferry Ride Cost starts at $ y = 1 $ when $ x = 0 $, then it's not direct. So the three correct ones are Breakfast Cost, Cost of Cinnamon, and Euro to U.S Dollar Conversion? Wait, no, the Breakfast Cost graph, Cost of Cinnamon graph, and Euro to U.S Dollar Conversion graph pass through the origin and have a constant ratio of $ y/x $.

Wait, the Breakfast Cost graph: points (1,2) and (6,12) – passes through (0,0). The ratio $ y/x = 2 $, so $ y = 2x $, direct variation.

Cost of Cinnamon: points (1,0.3) and (5,1.5) – passes through (0,0). Ratio $ y/x=0.3 $, so $ y = 0.3x $, direct variation.

Euro to U.S Dollar Conversion: points (1,1.4) and (5,7) – passes through (0,0). Ratio $ y/x = 1.4 $, so $ y = 1.4x $, direct variation.

So the three correct graphs are:

- Breakfast Cost

- Cost of Cinnamon

- Euro to U.S Dollar Conversion

(And also, maybe the Ferry Ride Cost was misread. Wait, the Ferry Ride Cost graph: when $ x = 1 $, $ y = 2 $; $ x = 5 $, $ y = 8 $. The slope is $ (8 - 2)/(5 - 1)=6/4 = 1.5 $, but the y - intercept: using $ y=mx + b $, $ 2=1.5(1)+b$, $ b = 0.5 $. So it has a y - intercept of 0.5, so not direct variation. The Parking Garage Rates: $ y=mx + 2 $, since when $ x = 0 $, $ y = 2 $, so not direct. So the three correct are Breakfast Cost, Cost of Cinnamon, and Euro to U.S Dollar Conversion.)

# Answer:
The three graphs that show functions with direct variation are:

- Breakfast Cost (the graph with "Number of Meals" on the x - axis, passing through the origin and points like (1,2), (6,12))

- Cost of Cinnamon (the graph with "Quantity (ounces)" on the x - axis, passing through the origin and points like (1,0.3), (5,1.5))

- Euro to U.S Dollar Conversion (the graph with "Euros" on the x - axis, passing through the origin and points like (1,1.4), (5,7))

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