Direct variation comes up often in math and is also used in our everyday lives. However, there are still lots of questions that you can ask about direct variation. Show
So, what is direct variation? Direct variation is a linear relationship between two variables x and y, where the ratio y / x is always equal to a constant value k. The equation has the form y = kx, and it has only two variables, each with exponents of 1. The graph of the equation is a line that passes through the origin (0, 0). Of course, the constant k in a direct variation can be negative or a fraction (or both). Both of these cases have a specific interpretation. In this article, we will talk about direct variation, what it is, and why it is important. We’ll also look at some examples of equations, tables, and graphs that show (or do not show) direct variation. Let’s get started. What Is Direct Variation?Direct variation is a relationship between two variables x and y where the ratio y / x is equal to a constant value k. We can write the equation:
or, solving for y:
Note that this is the equation of a line with a y-intercept of zero (zero constant term). In this case, we say that “y is directly proportional to x” or “y varies directly with x”. How To Find The Formula Of An Expon... Please enable JavaScript How To Find The Formula Of An Exponential FunctionThe value k is called the constant of variation. It tells us how much the value of y changes every time we change the value of x. For example, if we have the direct variation y = 2x, then y increases by 2 whenever x increases by 1. Similarly, y decreases by 2 whenever x decreases by 1. If we have the direct variation y = 10x, then y increase by 10 whenever x increases by 1. Similarly, y decreases by 10 whenever x decreases by 1. You can see an example of a table for the direct variation y = 2x below. xy0012243648This table shows thedirect variation relationship y = 2x. Why Is Direct Variation Important?Direct variation is important because it can be used in several applications in our everyday lives. For example, we can use direct variation for:
So, there are plenty of uses of direct variation in our lives. However, this still leaves the question of how to tell that an equation, table, or graph is a direct variation. How To Tell If An Equation Is A Direct VariationTo tell if an equation is a direct variation, look for the form y = kx. In particular, this means:
Example 1: An Equation That Is A Direct VariationConsider the equation:
This is a direct variation, since:
Example 2: An Equation That Is Not A Direct VariationConsider the equation:
This is not a direct variation, since:
Example 3: An Equation That Is Not A Direct VariationConsider the equation:
This is not a direct variation, since:
Example 4: An Equation That Is Not A Direct VariationConsider the equation:
This is not a direct variation, since:
Note: this is an example of inverse variation, since y = 2 / x. Example 5: An Equation That Is Not A Direct VariationConsider the equation:
This is not a direct variation, since:
How To Tell If A Table Is A Direct VariationTo tell if a table is a direct variation, look for a linear relationship between the columns in the table. In particular, this means:
Example 1: A Table That Is A Direct VariationConsider the table: xy00132639412This table shows thedirect variation relationship y = 3x. This table is a direct variation, since:
Example 2: A Table That Is Not A Direct VariationConsider the table: xyz00012527838941012This table shows arelationship between variables x, y, and z. This table is not a direct variation, since:
Example 3: A Table That Is Not A Direct VariationConsider the table: xy0113253646This table shows arelationship that is not direct variation. This table is not a direct variation, since:
Example 4: A Table That Is Not A Direct VariationConsider the table: xy01132639412This table shows thedirect variation relationship y = 3x. This table is not a direct variation, since:
How To Tell If A Graph Is A Direct VariationTo tell if a graph is a direct variation, look for a line that passes through the origin. In particular, this means:
Example 1: A Graph That Is A Direct VariationConsider the graph:
This graph is a direct variation, since:
Example 2: A Graph That Is Not A Direct VariationConsider the graph:
This graph is not a direct variation, since:
Example 3: A Graph That Is Not A Direct VariationConsider the graph:
This graph is not a direct variation, since:
How To Solve Direct VariationTo solve direct variation, plug in matching values of variables to find k. Then, if necessary, use the value of k with a value of x or y to find the other variable’s value. Example 1: Solving A Direct Variation For kLet’s say that there is a direct variation between x and y. That means:
for some constant k. Let’s assume we know that y = 15 when x = 3. Plugging in these values of x and y, we get:
So, the constant of variation is 5, and our equation is y = 5x. Example 2: Solving A Direct VariationLet’s say that there is a direct variation between x and y. That means:
for some constant k. Let’s assume we know that y = 12 when x = 2. What is y when x = 7? First, we need to find k. Plugging in x = 2, y = 12, we get:
So, the constant of variation is 6, and our equation is y = 6x. Now, we can find y when x = 7:
So, y = 42 when x = 7. Can Direct Variation Be Negative?Direct variation can be negative. To be more precise, the constant of variation k can be less than zero (k < 0). In practical terms, k < 0 means that y decreases as x increases. Similarly, y increases as x decreases. For example, if k = -5, then we have the direct variation y = -5x. When x increases by 1, y will decrease by 5. When x decreases by 1, y will increase by 5. The graph will be a line that goes through the origin (0, 0) and moves down as we move from left to right. The slope is -5 = -5 / 1 = rise / run, so we move down 5 units for every unit we move right.
Can Direct Variation Be A Fraction?Direct variation can be a fraction. To be more precise, the constant of variation k can be a rational number (or even an irrational number). In practical terms, if k is a fraction, we can interpret it as the slope (rise over run). So, if k is the fraction a / b, then we can write:
Another interpretation is that every time x increases by b, y increases by a. For example, if k = 2 / 3, then every time x increases by 3, y increases by 2. Every time x decreases by 3, y decreases by 2. Note that k can be both negative and a fraction. For example, we could have k = -5 / 4. Can A Parabola Be A Direct Variation?A parabola cannot be direct variation. Remember that the graph of a parabola comes from a quadratic equation. Every quadratic equation has a quadratic term which contains x2 (an exponent of 2). However, direct variation only applies to an equation y = kx, which has no x2 term. (You can learn more about parabolas in my article here.) Can Direct Variation Have Exponents?Direct variation cannot have exponents other than 1 for the variables x and y. This is because direct variation is a linear relationship between x and y. If the variables x and y have exponents other than 1, then we do not have a direct variation relationship. For example:
ConclusionNow you know what direct variation is and how to tell from an equation, table, or graph. You also have some examples that do not show direct variation between two variables. I hope you found this article helpful. If so, please share it with someone who can use the information. What is the slope in the equation y 4x 3?Algebra Examples
Using the slope-intercept form, the slope is 4 .
What is the slope of 4x Y =Using the slope-intercept form, the slope is −4 .
What equation represents slope intercept form?The slope intercept formula y = mx + b is used when you know the slope of the line to be examined and the point given is also the y intercept (0, b). In the formula, b represents the y value of the y intercept point.
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