Question

1. write an equation that represents the relationship between ( x ) milligrams of cayenne pepper and ( y ) milligrams of onion powder. show or explain your thinking.
2. graph the relationship on the coordinate plane.
3. how much onion powder is needed for 64 milligrams of cayenne pepper? show or explain your thinking.

(coordinate plane with onion powder (mg) on x - axis: 0, 10, 20, 30, 40, 50, 60, 70 and cayenne pepper (mg) on y - axis: 0, 10, 20, 30, 40, 50, 60)

Explanation:

Step1: Identify Relationship Type

Assume linear relationship (since graph is a line). Let's find two points. From the axes, when \( x = 20 \) (cayenne), \( y = 60 \) (onion)? Wait, no, labels: x is cayenne (mg), y is onion (mg)? Wait, the graph: x-axis (horizontal) is Onion Powder (mg), y-axis (vertical) is Cayenne Pepper (mg)? Wait, the labels: "Onion Powder (mg)" on horizontal (left to right: 10,20,30,40,50,60,70), "Cayenne Pepper (mg)" on vertical (0,10,20,30,40,50,60). Wait, the line passes through (20,60)? No, maybe (20,60) reversed? Wait, maybe the relationship is proportional. Let's check: if when cayenne is 20, onion is 60? Wait, no, let's re-express. Let's take two points: suppose when \( x = 20 \) (cayenne), \( y = 60 \) (onion)? Wait, no, the axes: horizontal is Onion (x), vertical is Cayenne (y). Wait, the line: when x=20 (onion), y=60 (cayenne)? No, the graph's line: let's see, the line crosses, maybe (20,60) and (30,90)? No, maybe it's a proportional relationship. Let's assume it's a linear equation \( y = kx \), where y is onion, x is cayenne. Wait, when x=20, y=60? Then \( k = 60/20 = 3 \). So equation: \( y = 3x \) (onion = 3 * cayenne). Let's verify: if cayenne is 20, onion is 60 (matches). If cayenne is 30, onion is 90, etc. So the relationship is linear, proportional, with slope 3.

Step2: Graph the Relationship

Plot points: when x=10 (cayenne), y=30 (onion); x=20, y=60; x=30, y=90 (but onion axis goes up to 70? Wait, maybe I mixed up axes. Wait, the horizontal axis is Onion (mg), vertical is Cayenne (mg). So x is onion, y is cayenne. Then when x=60 (onion), y=20 (cayenne). So slope \( k = 20/60 = 1/3 \). Ah, that makes sense! I had axes reversed. So x (onion) and y (cayenne). So points: (60,20) and (30,10). So slope \( m = (20 - 10)/(60 - 30) = 10/30 = 1/3 \)? No, (10, y) when x=30, y=10? Wait, no, let's take two points: when onion (x) is 60, cayenne (y) is 20; when onion is 30, cayenne is 10. So slope \( m = (20 - 10)/(60 - 30) = 10/30 = 1/3 \)? No, \( m = (y2 - y1)/(x2 - x1) = (10 - 20)/(30 - 60) = (-10)/(-30) = 1/3 \). So equation: \( y - 20 = (1/3)(x - 60) \). Simplify: \( y = (1/3)x - 20 + 20 \)? Wait, \( y = (1/3)x \). Wait, when x=60, y=20: \( 20 = (1/3)(60) = 20 \), correct. When x=30, y=10: \( 10 = (1/3)(30) = 10 \), correct. So the equation is \( y = \frac{1}{3}x \), where y is cayenne (mg), x is onion (mg). So cayenne = (1/3) * onion, or onion = 3 * cayenne. Wait, that's the same as before, just reversed. So if we let x be cayenne, y be onion: \( y = 3x \). If x is onion, y is cayenne: \( y = (1/3)x \). Let's confirm with the graph: when onion (x) is 60, cayenne (y) is 20: \( 20 = (1/3)(60) \), yes. When onion is 30, cayenne is 10: \( 10 = (1/3)(30) \), yes. So the correct equation, depending on variables: let's define x = onion powder (mg), y = cayenne pepper (mg). Then \( y = \frac{1}{3}x \), or \( x = 3y \).

Step3: Solve for Onion when Cayenne is 64

We need to find x (onion) when y (cayenne) is 64. Using \( x = 3y \) (since onion = 3 * cayenne). So \( x = 3 * 64 = 192 \)? Wait, no, wait: if \( y = \frac{1}{3}x \), then \( x = 3y \). So when y=64, x=3*64=192. Wait, but the onion axis only goes up to 70. That can't be. Wait, I must have reversed the variables. Let's re-express: the graph's horizontal axis is Cayenne Pepper (mg), vertical is Onion Powder (mg). So x (horizontal) is cayenne, y (vertical) is onion. Then when x=20 (cayenne), y=60 (onion); x=30, y=90 (onion). So equation: \( y = 3x \) (onion = 3 * cayenne). Then when cayenne is 64, onion is 3*64=192. But the onion axis on the graph goes up to 70, but maybe the graph is a part of the line. So the equation is \( y = 3x \), where y is onion (mg), x is cayenne (mg).

So for question 4: How much onion powder is needed for 64 mg cayenne? Use \( y = 3x \), x=64, so y=3*64=192 mg.

Answer:

For equation: \( y = 3x \) (where \( y \) is onion powder in mg, \( x \) is cayenne pepper in mg).
For graph: Plot points (10,30), (20,60), (30,90), etc., and draw a line through them.
For question 4: 192 milligrams of onion powder.