Question

7 the equation for a proportional relationship is ( y = 5.8x ). the graph of the relationship passes through the point ( (1, 5.8) ) which represents the ______. (handwritten: the constant of proportionality)
8 mindy says that the equations ( p = 1.5q ) and ( \frac{2}{3}p = q ) both represent the same proportional relationship. vivian says that cannot be true because the constants of proportionality are different. which student is correct? explain.
9 math journal graph a proportional relationship. then write an equation that represents the proportional relationship. explain how to find the constant of proportionality in both your equation and your graph. (with a grid graph)

Explanation:

Question 7

Step1: Recall proportional relationship formula

The general form of a proportional relationship is \( y = kx \), where \( k \) is the constant of proportionality. When \( x = 1 \), \( y = k\times1 = k \).

Step2: Analyze the given point

For the equation \( y = 5.8x \), when \( x = 1 \), \( y = 5.8\times1 = 5.8 \). So the point \( (1, 5.8) \) has \( x = 1 \) and \( y = 5.8 \), which means \( k = 5.8 \) here. This point represents the constant of proportionality because in \( y = kx \), when \( x = 1 \), \( y = k \).

Step1: Recall the form of proportional relationship

A proportional relationship can be written as \( y = kx \) (or in terms of \( p \) and \( q \), \( p = kq \) or \( q = kp \), depending on which variable is dependent/independent).

Step2: Transform the second equation

Start with \( \frac{2}{3}p = q \). Multiply both sides by \( \frac{3}{2} \) to solve for \( p \). So \( p=\frac{3}{2}q \), and \( \frac{3}{2}=1.5 \).

Step3: Compare the two equations

The first equation is \( p = 1.5q \) and the second, after transformation, is also \( p = 1.5q \). So they represent the same proportional relationship. The "different" constants of proportionality idea is wrong because the constants depend on which variable is considered the dependent one. If we write \( p \) in terms of \( q \), the constant for \( p = kq \) is \( 1.5 \) in both cases (after transforming the second equation). So Mindy is correct.

Step1: Choose a constant of proportionality

Let's choose \( k = 2 \). So the equation of the proportional relationship is \( y = 2x \).

Step2: Graph the relationship

• For a proportional relationship \( y = kx \), the graph is a straight line passing through the origin \( (0,0) \).
• Find some points: when \( x = 0 \), \( y = 0 \); when \( x = 1 \), \( y = 2\times1 = 2 \); when \( x = 2 \), \( y = 2\times2 = 4 \); when \( x = 3 \), \( y = 2\times3 = 6 \).
• Plot the points \( (0,0) \), \( (1,2) \), \( (2,4) \), \( (3,6) \) and draw a straight line through them.

Step3: Find constant of proportionality from equation

In the equation \( y = 2x \), the form is \( y = kx \), so the constant of proportionality \( k \) is the coefficient of \( x \), which is \( 2 \).

Step4: Find constant of proportionality from graph

For a proportional relationship graph (a straight line through the origin), the constant of proportionality \( k \) is the slope of the line. The slope \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Using two points, say \( (1,2) \) and \( (2,4) \), \( m=\frac{4 - 2}{2 - 1}=\frac{2}{1}=2 \). Also, for any point \( (x,y) \) on the line (other than the origin), \( k=\frac{y}{x} \). For \( (1,2) \), \( \frac{y}{x}=\frac{2}{1}=2 \); for \( (2,4) \), \( \frac{4}{2}=2 \), etc.

Answer:

constant of proportionality

Question 8