What Does This Mean?
When you enter the world of high school math, you will be introduced to a number of unique equations. In addition, you learn how to graph them and find solutions for their respective equations. However, one of the biggest takeaways is that as the degree of the equation (highest exponent in the equation) increases, the difference which will stay the same also increases (Ex: Linear has first differences same, quadratic has second differences same, cubic has third differences same and so on). To put this into perspective, let’s look at a linear, quadratic, and a cubic relation in depth:
Thus, a linear relation has a straight line, which can be seen by the red line representing y = x. Moreover, a quadratic relation has a u-shaped graph, which can be seen by the blue line representing y = x^2. Lastly, a cubic relation is like a parabola, where the general idea is that half of the graph is reflected across the x-axis, which can be seen by the green line representing y = x^3. Now, let’s take a deep dive into these equations and derive exactly why the first, second and third differences are the same in the respective equations.
Linear Relations Analysis:
Linear Relations: The equation is defined by y = mx + b. Now, let’s take a look at the values of our line y = x.
As a result, the first differences in the table of values are the same. However, why is this the case? To answer that question, let’s analyze the equation by plugging in the various values of x and then see if we can understand it better.
Thus, that is why your math teacher tells you to find the first difference, as it will give you the slope or rate of change of the equation. Also, recall that this works because we have consecutive points, but if you have (2,y) and (4,y), the difference is not m, it is actually 2m –> so do not get confused. Not to mention, this is why our slope formula (y2-y1/x2-x1) works because that gives us m as we take the difference in the y values and then divide that by the difference in the x values (and to make sure that the formula applies to all coordinates, not just consecutive ones). Now, let’s move on to quadratic relations.
Quadratic Relations Analysis:
Quadratic Relations: The equation is defined by y = ax^2 + bx + c. Now, let’s take a look at the values of our line y = x^2.
As a result, the second differences in the table of values are the same. However, why is this the case? To answer that question, let’s once again analyze the equation by plugging in the various values of x and then see if we can understand it better.
Therefore, this is precisely why we look at the differences of an equation to determine whether it is quadratic. Not to mention, this is exactly why you may have noticed that the a value of the equation turns out to be half of the second difference in the table of values. In addition, using this information, we can create an equation for the value of a: a = (y2 – y1) / 2(x2 – x1). Note that this will only work with the first difference values of y. Now, let’s test it. We will use points 1, 1 (a + b) and 4, 7 (7a + b) from our original table of values – under the first difference section: a = (7-1) / 2(4-1) = 6/2(3) = 6/6 = 1. Hence, a is equal to 1, which makes sense because our equation was y = x^2 or y = 1x^2. Lastly, let’s explore cubic relations.
Cubic Relations Analysis:
Cubic Relations: The equation is defined by y = ax^3 + bx^2 + cx + d. Now, let’s take a look at the values of our line y = x^3.
As a result, the third differences in the table of values are the same. However, why is this the case? To answer that question, let’s again (for the third time) analyze the equation by plugging in the various values of x and then see if we can understand it better.
In conclusion, this is why we always examine the third differences in a table of values to check whether the equation is of the form of a cubic relation. Not to mention, by looking at the second difference y values, we can create an equation to find a: a = (y2-y1) / 6(x2-x1). Now, let’s test this. We will use the points 2,6 (6a + 2b) and 5,24 (24a + 2b) *6 and 24 are the second differences: a = (24-6) / 6(5-2) = 18/6(3) = 18/18 = 1. Alas, a is equal to 1, which makes sense because our equation was y = x^3 or 1x^3.
Overall, this logic is great, but what am I getting out of it? Well, we can use this intuition to find the equation of various sorts (ex: linear, quadratic or cubic), by creating equations and solving for the variables. Of course, let’s go through some examples to drive this point home.
Solve Problems Using This Logic:
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