正比：曲线、求斜率

Directly Proportional Graphs

Let's take a look at some directly proportional graphs when $k=2$, $k=\frac12$, and $k=6$.

Notice anything? They're all linear! And they all pass through the origin. So any linear equation that passes through the origin is a direct proportion.

If we write the above equations out, we get: $$y={\color{#4079b2}{\frac12}} x\quad\quad y={\color{#bf4f4c}{2}}x\quad\quad y={\color{#419d51}{6}}x$$ The constant of proportionality is just the slope of the linear equation.

Example

Suppose we're told to draw the graph of a proportion whose constant of proportionality is $10$. Can we?

Yes! Remember, the constant of proportionality is just the slope of a line. The equation is: $$y=10x$$

Graphing this gives us:

Practice Problem

A graph is given showing that the amount of money you make is directly proportional to the hours you worked.

Solution

We're told this graph is directly proportional so we know its equation is of the form: $$y=kx$$

Looking at the graph, we need to pick any $(x,y)$ coordinate and plug it in to solve for $k$. One noticeable point is $(30,540)$. $$\begin{align} y&=kx\\ 540&=k(30)\\ \frac{540}{\color{#8925ae}{30}}&=\frac{30k}{\color{#8925ae}{30}}\\ 18&=k \end{align}$$

This is our answer! You make $18\text{ dollars per hour}$.

Another way we can think of solving this problem is that we are finding the slope from the graph.

We can also show this by plugging $k$ in and finding the amount you make for working $1\text{ hour}$: $$\begin{align} y&=18x\\ y&=18(1)\\ y&=18 \end{align}$$ You make $18\text{ dollars}$ every hour.

(Video) Learn and Understand Direct Variation

by AlgebraTestHelper

This video by AlgebraTestHelper goes over how to do a direct variation problem.

Direct Variation

In algebra, we will see direct variation, inverse variation, and joint variation. This lesson will cover direct variation.

Here is the standard equation for direct variaiton:

$$y=kx$$

Where $y$ and $x$ vary directly. $k$ is the constant of variation.

For example, think of boiling water on a stove. As the pot is on the flame, the temperature is increasing, but so is the pressure in the pot. Temperature and pressure vary directly. As temperature increases, pressure increases.

Let's look at an example.

Example 1: If $x=3$ and $y=15$, what is $y$ when $x=5$?

First, let's solve for $k$.

$$y=kx$$ $$15=k(3)$$ $$5=k$$

So we know that the constant of variation, $k$, is $5$. So now, instead of having a general equation, we have a new equation for this relationship:

$$y=5x$$

What is $y$ when $x=5$? Let's take a look:

$$y=5(5)$$ $$y=25$$

Most direct variation problems will work out pretty similar to this example!

Direct Variation

We can use direct variation to describe a specific type of relationship between two variables.

Say we wanted to track the number of people who go to the beach and the temperature outside. If we sat on a park bench and counted the number of people who go to the beach everyday for one year, and we tracked the temperature for one year, we may notice that as the temperature increases, so does the number of people who go to the beach. This is an example of direct variation.

If we want to mathematically graph these numbers, there's an equation for it!