quadratic Summary
Quadratic Explanation:
In math, we reviewed and discussed many topics, which included the standard form of a quadratic, the coefficients a, b, and c, the vertex form of a quadratic, x-intercepts, and how to convert from standard form to vertex form and back. All of this together explains and describes the first lessons in math that we had as tenth grader.
In math, we reviewed and discussed many topics, which included the standard form of a quadratic, the coefficients a, b, and c, the vertex form of a quadratic, x-intercepts, and how to convert from standard form to vertex form and back. All of this together explains and describes the first lessons in math that we had as tenth grader.
Standard Form of a Quadratic and the Effects of the Coefficients to the Graph:
y=ax^2+bx+c
[x^2 presents the same as x squared throughout the whole page]
Coefficients- a, b
Constant- c
This standard form only applies to parabolas, a symmetrical curve defining the pattern being used, that are or have been graphed. The a, b, and c in the standard form are used as representatives for numbers, and if the numbers were changed, whether from positive to negative or to halved or doubled, it would change the shape of the parabola. If the coefficient a became a negative, the parabola would be flipped, the parabola becoming a maximum instead of a minimum. If the coefficient b became a negative, the parabola would be mirrored along the y-axis, the point number would be the same, except they would be the opposite of what they were before. If the constant c became a negative, the parabola would be lowered, until one of the points in the parabola intersected with the negative c on the y-axis. This is what would happen if the coefficients and constant were changed from a positive to a negative. However, the parabola can also be changed by changing the numerical value of a in the equation. As in, if the numerical value of a in the equation was greater than 1, the parabola would become thinner, if he numerical value of a in the equation was less than 1, the parabola would become wider. (Examples are shown below)
y=ax^2+bx+c
[x^2 presents the same as x squared throughout the whole page]
Coefficients- a, b
Constant- c
This standard form only applies to parabolas, a symmetrical curve defining the pattern being used, that are or have been graphed. The a, b, and c in the standard form are used as representatives for numbers, and if the numbers were changed, whether from positive to negative or to halved or doubled, it would change the shape of the parabola. If the coefficient a became a negative, the parabola would be flipped, the parabola becoming a maximum instead of a minimum. If the coefficient b became a negative, the parabola would be mirrored along the y-axis, the point number would be the same, except they would be the opposite of what they were before. If the constant c became a negative, the parabola would be lowered, until one of the points in the parabola intersected with the negative c on the y-axis. This is what would happen if the coefficients and constant were changed from a positive to a negative. However, the parabola can also be changed by changing the numerical value of a in the equation. As in, if the numerical value of a in the equation was greater than 1, the parabola would become thinner, if he numerical value of a in the equation was less than 1, the parabola would become wider. (Examples are shown below)
Vertex Form of a Quadratic:
The vertex of a parabola is the highest point, if it is a maximum parabola, or the highest point, if it is a minimum parabola (the parabola on the right is a minimum). To find out the vertex of a parabola, one could test all possible points until they start repeating. This, however, will be quite troublesome with number that are exponential or ones that are followed by many decimals, so there is a certain formula to figure it out easily, the vertex form. The vertex form is y=a(x-h)^2+k. The coefficients h and k represent the coordinates of the vertex, h is the x coordinate point, just switched, k is the y coordinate point. The a coefficient determines whether it is a maximum or minimum parabola, as spoken about before. For example, if the vertex form was y=2(x-3)^2+4, the vertex would be the coordinate point (3,4) as well as the parabola being a minimum. If we were looking at y=-5(x+2)^2-1, the vertex would be the coordinate point (-2,-1) as well as the parabola being a maximum, the reason being is that the replacement for the coefficient a is a negative number, hence the parabola being a maximum. The coordinate points are negative because the opposite of a positive two is a negative two, and the negative one automatically becomes the y coordinate for the point.
The vertex of a parabola is the highest point, if it is a maximum parabola, or the highest point, if it is a minimum parabola (the parabola on the right is a minimum). To find out the vertex of a parabola, one could test all possible points until they start repeating. This, however, will be quite troublesome with number that are exponential or ones that are followed by many decimals, so there is a certain formula to figure it out easily, the vertex form. The vertex form is y=a(x-h)^2+k. The coefficients h and k represent the coordinates of the vertex, h is the x coordinate point, just switched, k is the y coordinate point. The a coefficient determines whether it is a maximum or minimum parabola, as spoken about before. For example, if the vertex form was y=2(x-3)^2+4, the vertex would be the coordinate point (3,4) as well as the parabola being a minimum. If we were looking at y=-5(x+2)^2-1, the vertex would be the coordinate point (-2,-1) as well as the parabola being a maximum, the reason being is that the replacement for the coefficient a is a negative number, hence the parabola being a maximum. The coordinate points are negative because the opposite of a positive two is a negative two, and the negative one automatically becomes the y coordinate for the point.
Changing Vertex Form to Standard Form and Back:
Vertex form is y=a(x-h)^2+k, to convert this to standard form, one just needs to write out the problem. For example, take
y=2(x-3)^2+4 Write out the problem, or get rid of the exponent sign and solve through
y=2(x-3)(x-3)+4 Here, we took away the ^2 and replaced it with the term in parenthesis
y=2(x^2-3x-3x+9)+4 As shown to the left, we multiplied the parenthesis together to get this
y=2(x^2-6x+9)+4 The next step was to combine the like terms in the parenthesis
y=2x^2-12x+18+4 The 2 was then distributed to the terms in the parenthesis
y=2x^2-12x+22 Finally, we combine the like terms, which give us the standard form
Standard form is y=ax^2+bx+c, to convert to vertex form, one needs to factor out the equation given. For example, take
y=-1(5x^2+20x+25) The first thing we would need to do is divide each term by 5 to make
y=-5(x^2+4x+5) the coefficient a equal to one
4= 1+3 or 2+2 We find different factors that add up to 4, and mark the one that repeats
y=-5(x+2)^2 +5 The +5 is on the side, and the (x+2)^2 in the middle, to show the (x-h)^2
2x2=4 -4 We multiply the 2 in parenthesis by itself, add the opposite to the k term
y=-5(x+2)^2+1 That is how you transfer standard form to vertex form
Tips:
Vertex form is y=a(x-h)^2+k, to convert this to standard form, one just needs to write out the problem. For example, take
y=2(x-3)^2+4 Write out the problem, or get rid of the exponent sign and solve through
y=2(x-3)(x-3)+4 Here, we took away the ^2 and replaced it with the term in parenthesis
y=2(x^2-3x-3x+9)+4 As shown to the left, we multiplied the parenthesis together to get this
y=2(x^2-6x+9)+4 The next step was to combine the like terms in the parenthesis
y=2x^2-12x+18+4 The 2 was then distributed to the terms in the parenthesis
y=2x^2-12x+22 Finally, we combine the like terms, which give us the standard form
Standard form is y=ax^2+bx+c, to convert to vertex form, one needs to factor out the equation given. For example, take
y=-1(5x^2+20x+25) The first thing we would need to do is divide each term by 5 to make
y=-5(x^2+4x+5) the coefficient a equal to one
4= 1+3 or 2+2 We find different factors that add up to 4, and mark the one that repeats
y=-5(x+2)^2 +5 The +5 is on the side, and the (x+2)^2 in the middle, to show the (x-h)^2
2x2=4 -4 We multiply the 2 in parenthesis by itself, add the opposite to the k term
y=-5(x+2)^2+1 That is how you transfer standard form to vertex form
Tips:
- The term h can be found if you divide the b coefficient by two (h=b/2)
- The term k can be found if you subtract h squared from c (k=c-h^2)
X-Intercepts or Roots:
The x intercept and the y intercept aren't that different when talking about everyday math, one is horizontal, the other vertical. A line, whether curved or straight, usually passes both of these lines at some point. However, with parabolas, the chances of the curved line passing the x axis isn't always certain. This is because a parabola has three different 'options', to cross the x axis twice, once, or not at all. A parabola crosses the x axis twice on occasion, meaning that a maximum parabola is in quadrants I and II of the graph, or if it's a minimum, quadrant III and IV. A parabola that only crosses the x axis once means than the vertex of the parabola is on the x axis, and it can be a minimum or a maximum. For a parabola to not cross the x axis at all means that either a maximum parabola is in quadrant III or IV or that a minimum parabola is in quadrant I or II.
The x intercept and the y intercept aren't that different when talking about everyday math, one is horizontal, the other vertical. A line, whether curved or straight, usually passes both of these lines at some point. However, with parabolas, the chances of the curved line passing the x axis isn't always certain. This is because a parabola has three different 'options', to cross the x axis twice, once, or not at all. A parabola crosses the x axis twice on occasion, meaning that a maximum parabola is in quadrants I and II of the graph, or if it's a minimum, quadrant III and IV. A parabola that only crosses the x axis once means than the vertex of the parabola is on the x axis, and it can be a minimum or a maximum. For a parabola to not cross the x axis at all means that either a maximum parabola is in quadrant III or IV or that a minimum parabola is in quadrant I or II.
Personal Growth-
During this unit, I became less of a talker and more of a listener, I usually dominated the conversation and when my group mates didn't say anything I just continued to talk.Now, however, I began to try and get other people at the table to talk more and I would listen to what they had to say and stated my two cents of information. I also began to organize my math worksheets more, before I had a problem keeping track of all the work I did, but when we started referring back to the work I began to organize myself more. Within our groups I continued to show these traits by organizing our group work that we would need to turn in, I would help my teammates with the problem if they needed it, I would show my teammates my way of solving the problem and explain how it would work, and when they were talking I make sure to pay attention and take note of what they are saying. This was how I showed collaborating & listening and staying organized while working with my groups.
During this unit, I became less of a talker and more of a listener, I usually dominated the conversation and when my group mates didn't say anything I just continued to talk.Now, however, I began to try and get other people at the table to talk more and I would listen to what they had to say and stated my two cents of information. I also began to organize my math worksheets more, before I had a problem keeping track of all the work I did, but when we started referring back to the work I began to organize myself more. Within our groups I continued to show these traits by organizing our group work that we would need to turn in, I would help my teammates with the problem if they needed it, I would show my teammates my way of solving the problem and explain how it would work, and when they were talking I make sure to pay attention and take note of what they are saying. This was how I showed collaborating & listening and staying organized while working with my groups.