Show a proportional graph and equation and compare to a non proportional graph and equation.
Proportional: y =
Non Proportional: y =
To illustrate a proportional graph and equation, let's consider the equation y = 2x. This equation represents a direct proportion, where y is always twice the value of x.
Here is a table of values for the proportional equation:
| x | y |
|-------|-------|
| 1 | 2 |
| 2 | 4 |
| 3 | 6 |
| 4 | 8 |
| 5 | 10 |
Now, let's plot these points on a graph:
graph(x,2x)
This graph shows a straight line that passes through the origin (0,0) and has a positive slope of 2.
For the non-proportional graph and equation, let's consider the equation y = x^2. This equation represents a non-proportional relationship, where y is not directly proportional to x.
Here is a table of values for the non-proportional equation:
| x | y |
|-------|-------|
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |
| 5 | 25 |
Now, let's plot these points on a graph:
graph(x,x^2)
This graph shows a curve, often referred to as a parabola. The y-values do not change at a constant rate as x increases, indicating a non-proportional relationship.
In summary, a proportional equation and graph (y = 2x) represents a direct proportion, where the ratio of y to x is constant. In contrast, a non-proportional equation and graph (y = x^2) does not have a constant ratio between y and x, representing a non-proportional relationship.
Proportional:
A proportional graph represents a direct relationship between two variables. The equation for a proportional graph is in the form of y = mx, where m is the slope of the line.
For example, let's consider the equation y = 2x. This equation represents a straight line that passes through the origin (0,0) with a slope of 2. It indicates that as the value of x increases, the value of y also increases in a consistent proportion. This graph would be an example of a proportional relationship.
Non-Proportional:
A non-proportional graph, on the other hand, does not show a direct relationship between the variables. The equation for a non-proportional graph is generally not in the form y = mx.
For example, let's consider the equation y = x^2. This equation represents a parabola opening upwards. As the value of x increases, the value of y does not increase at a consistent proportion. Instead, it gradually increases, and the rate of change would vary at different points on the graph. This graph would be an example of a non-proportional relationship.
To summarize, a proportional graph represents a direct relationship between variables where the ratio of y to x remains constant. A non-proportional graph, on the other hand, does not exhibit a constant ratio between y and x, resulting in a more complex relationship.
To show a proportional graph and equation, we need to understand what it means for a relationship to be proportional. In a proportional relationship, as one quantity increases or decreases, the other quantity also increases or decreases by the same factor. This means that the graph of a proportional relationship will always form a straight line passing through the origin (0,0).
The equation for a proportional relationship can be written as:
y = kx
Where:
- "y" represents the dependent variable
- "x" represents the independent variable
- "k" represents the constant of proportionality
In this equation, the value of "k" determines the rate at which the dependent variable changes with respect to the independent variable. It remains constant throughout the relationship.
As an example, let's consider a proportional relationship where y represents the total cost of buying x items at a constant price per item. Let's assume the price is $5 per item, and if we buy 2 items, the total cost would be:
y = 5x
So, for 2 items:
y = 5 * 2
y = 10
For a non-proportional graph and equation, the relationship between the variables does not follow the proportional pattern. In a non-proportional relationship, the dependent variable does not change by a constant factor as the independent variable changes.
Let's take an equation as an example:
y = x^2
In this equation, the dependent variable "y" is not directly proportional to the independent variable "x." As "x" increases or decreases, "y" changes at an increasing rate based on the square of "x." The graph would not form a straight line passing through the origin.
To summarize, in a proportional graph and equation, the relationship between the variables can be expressed using a linear equation with a constant of proportionality. On the other hand, a non-proportional graph and equation show a relationship where the dependent variable does not vary proportionally with the independent variable.