Question

lesson 2 - 4
describe proportional relationships: constant of proportionality
quick review
the equation ( y = kx ) describes a proportional relationship between two quantities ( x ) and ( y ), where ( k ) is the constant of proportionality. ( k=\frac{y}{x} ) for any related pair of ( x ) and ( y ) except when ( x = 0 ).
example
the table shows the wages roger earned for the hours he worked. what equation relates the wages, ( w ), and the number of hours, ( h )? find the constant of proportionality, ( k ).
( \frac{27}{3}=9 ) ( \frac{45}{5}=9 ) ( \frac{54}{6}=9 )
write the equation in the form ( y = kx ).
( w = 9h )
practice

1. sally is going on vacation with her family. in 2 hours they travel 90.5 miles. if they travel at the same speed, write an equation that represents how far they will travel, ( d ), in ( h ) hours.
2. the table shows the weights of bunches of bananas and the price of each bunch. identify the constant of proportionality. write an equation to relate weight, ( w ), to the price, ( p ).

lesson 2 - 5
graph proportional relationships
quick review
the graph of a proportional relationship is a straight line through the origin. you can identify the constant of proportionality, ( k ), from the point ( (1,k) ) or by dividing ( \frac{y}{x} ) for any point ( (x,y) ) except the origin.
example
is the number of sunny days proportional to the number of rainy days? if so, find the constant of proportionality, and explain its meaning in this situation.
the graph is a straight line through the origin, so it shows a proportional relationship. the constant of proportionality is 2, which means for every 1 rainy day, there were 2 sunny days.
practice

1. does the graph show a proportional relationship? explain.
2. sketch a graph that represents a proportional relationship.

Explanation:

Lesson 2 - 4

Practice 1

Step1: Find the speed (constant of proportionality)

The speed \(k\) is calculated by dividing the distance \(d\) by the time \(h\). Given \(d = 90.5\) miles and \(h=2\) hours. So, \(k=\frac{d}{h}=\frac{90.5}{2}=45.25\)

Step2: Write the equation

Using the form \(y = kx\) (here \(d\) is like \(y\) and \(h\) is like \(x\)), the equation is \(d = 45.25h\)

Step1: Find the constant of proportionality \(k\)

For the first pair: \(k_1=\frac{p}{w}=\frac{1.35}{3}=0.45\)
For the second pair: \(k_2=\frac{p}{w}=\frac{1.71}{3.8}=0.45\)
For the third pair: \(k_3=\frac{p}{w}=\frac{2.34}{5.2}=0.45\)

Step2: Write the equation

Using the form \(y = kx\) (here \(p\) is like \(y\) and \(w\) is like \(x\)), the equation is \(p=0.45w\)

The graph of a proportional relationship is a straight - line through the origin. In the given "County Fair" graph, it is a straight line and passes through the origin \((0,0)\). We can also check the constant of proportionality. Let's take a point \((4,4)\) (assuming from the graph's scale, if \(x = 4\) (number of tickets) and \(y = 4\) (cost)). Then \(k=\frac{y}{x}=\frac{4}{4}=1\). For another point, say if \(x = 8\) and \(y = 8\) (from the line), \(k=\frac{y}{x}=\frac{8}{8}=1\). Since it is a straight line through the origin and the ratio \(\frac{y}{x}\) (constant of proportionality) is the same for non - origin points.

Answer:

\(d = 45.25h\)

Practice 2