PLEASE REPLY TO EACH DISCUSSION WITH SIGNIFICANT CONTENT. THERE ARE TWO DISCUSSIONS. DO NOT JUST AGREE WITH THE STUDENT BUT PROVIDE SUPPORTING CONTENT IF YOU AGREE OR DISAGREE. THERE ARE TWO DISCUSSION RESPONSE. PLEASE RESPOND WITH A MINUMUM OF 100 words FOR EACH DISCUSSION.
Week One Think and Answer
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1a. Linear equations and Linear Inequalities share very similar properties in approaching how to solve them. For example, the Addition and Subtraction Property of Equality for Linear Equations states that, “adding the same expression to each side of an equation or subtracting the same expression from each side of an equation produces an equivalent equation” (Aufmann, 76). Then the Addition-Subtraction Property for Linear Inequalities states, “If the same real number is added to or subtracted from each side of an inequality, the resulting inequality is equivalent to the original inequality” (Aufmann, 121). Essentially, for both, adding or subtracting the same value to both sides of an equation or inequality results in an equivalent equation or inequality. They also both seem to require isolating whatever given variable. One major difference I notice between Linear Equations and Linear Inequalities is in their Multiplication/Division properties. In Linear Inequalities, “multiplying or dividing each side of an inequality by the same positive real number produces an equivalent inequality” (Aufmann, 121), and “multiplying or dividing each side of an inequality by the same negative real number produces an equivalent inequality provided the direction of the inequality symbol is reversed” (Aufmann, 121). Whereas, in Linear Equations, simply multiplying or dividing each side by the same expression, positive or negative, results in an equivalent equation.
2a. A relation is defined by “any set of ordered pairs” (Aufmann, 164), while a function “is a set of ordered pairs in which no two ordered pairs have the same first coordinate and different second coordinates” (Aufmann, 165). So even though every function is a relation, not every relation can be considered a function because, by its definition, a function has specific requirements of an ordered pair. Whereas, a relation is literally any set of ordered pairs.
3a. The Fundamental Theorem of Algebra, proven by mathematician Carl Friedrich Gauss, states that “if P is a polynomial function of degree n is greater than or equal to 1 with complex coefficients, then P has at least one complex zero” (Aufmann, 299). This is significant because with this theorem, Gauss was the first to prove that every polynomial function has at least one complex zero and it is a foundational concept in studying algebra. It also has led to the development of other theorems like the Linear Factor Theorem. The Linear Factor Theorem along with the Number of Zeros of a Polynomial Function Theorem are considered existence theorems. These are important because “they state that an nth-degree polynomial will have n linear factors and n complex zeros” (Aufmann, 300). However, these theorems do not explain the how of determining the linear factors or zeros.
Aufmann, Richard, and Richard Nation. Algebra & Trigonometry. Eighth Edition. Stamford: Cengage Learning, 2015. Print.
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Week 1 Discussion
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1a. Linear equations and inequalities can both be shown via graph or problem. They both can be solved in the same fashion like graphing, substitution and elimination. However, the main difference is how they are computed. A linear equation has only a specific answer. Ex: X+2=8 where X can only be 6. But with Linear inequalities, you can have multiple answers. Ex: X+2<8, now X can be any number less than 8, so 7 and below can solve this problem. On a graph, a Linear Equation is represented by dots with a line that can go through them depending on the problem you are working on. Whereas an inequality will have a start point and either go on indefinitely depending on the direction of the sign. A good example is when they are shown on a graph. Inequalities moreover, will have shaded regions, which are the areas that can solve the problem; a linear equation will only have a line.
2a. Not all relations can be functions. A relation is a set of ordered pairs. (2,6) the 2 is called the domain and the 6 is called the range. You can not have 2 same domains in a function. (2,6) (3,8) (2,9), this is not a function since it has 2 domains. A good way to test this out is on a graph. This test is called the vertical line test which if 2 x values fall on the vertical line, then it fails as a function. A good function is (2,6) (6,8) (3,8). A good function equation is y=x2, in that equation the vertical line test passes since the function on a graph turns into a U(ish) shape.
3b. A rational function has asymptote because it is a hypothetical position that a rational function can approach but never touch. A rational function is consisted of two polynomial functions where the denominator is no equal to 0. The reason why a rational function has asymptote while the polynomial function does not is because if you do a horizontal/vertical line test, a rational function will not cross twice. Also since the denominator can not be zero, when you find the singularities, those points are where the asymptotes are, since that is the point that the rational function can not cross. It will go for infinity. A polynomial function can cross the vertical/horizontal line test multiple times. Take a look at a cubic polynomial function which can cross the horizontal line more than once, depending on the problem you are solving. Also a polynomial can equal to zero to solve to problem as well.
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