The Journey to Learn Quadratics
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Table of contents
- Characteristics of a Parabola- Second Differences & Quadratic Relations
Also, in this course, we learned that Quadratic equations and relations can come in 3 different forms:
Types of Quadratic Equations
- Vertex form
- Factored form
- Standard form
- Determining an equation when given the vertex
- Isolating for x
- Graphing vertex form
- Graphing factored form
- Common factoring
- Binomial common factoring
- Factoring by grouping
- Perfect squares
- Difference of squares
- Factoring simple trinomials
- Factoring complex trinomials
- Completing the square
- Using the Quadratic Formula
- Graphing standard form
- Motion Problems
- Revenue Problems
- Shape Problems
- Area Problems
Structure of a Parabola
There is much more to a Parabola other than just a line, like linear relations. A parabola is the shape a quadratic relation takes when it is graphed.
The Vertex is the maximum of minimum point on the graph. For parabola's facing upward, the vertex is the minimum point and for parabola's facing downward, the vertex is the maximum point.The vertex is the location where the parabola starts to change its direction.
The Optimal Value is also known as the maximum/minimum value. For parabola's facing upward, the optimal value is the minimum value and for parabola's facing downward, the optimal value is the maximum value. The y-coordinate of the vertex is always the same as the optimal value. To find out the optimal value, look at the y-value of the vertex.
The Axis of Symmetry of a parabola is a vertical line that cuts the parabola into two equal halves. The x-coordinate of the vertex is always the same as the axis of symmetry. the axis of symmetry is located halfway between the zeroes, if there are any.
Direction of Opening: The direction of opening of a parabola is the way the parabola is facing. A parabola can open in 2 directions; upwards or downwards. In any form( standard, factored, vertex), a parabola facing downwards would have a negative a value. A parabola facing upwards would have a positive a value.
Possible x-values for any parabola will always be a set of real numbers.This is due to the fact that all x-values are possible on any parabola ( if you extend the arms all the way, then all the real numbers on the x-axis are possible x-values).
Possible y values depends on the optimal value.These are the possibilities based on the optimal value and if the optimal value is a maximum or minimum. So if the optimal value is a maximum (opening downwards), then the possible y-values are all real numbers less than the optimal value (possible y-value <optimal value). If the optimal value is a minimum( opening upwards), then the possible y-values are all real numbers greater than the optimal value (possible y-values > optimal value).
Example: Lets find the direction of opening,vertex, axis of symmetry, optimal value, possible x-values and possible y-values for the parabola with the equation:Y=-(x-4)^2+12
I know that the a-value is -1 the -1 in this example is in the same place as the a value in because in the basic vertex form equation (look at section: types of equations) Y=a(x-h)^2+k h.
Just by looking at the a value, I know that the parabola will be facing downwards. I know this because The a value( leading coefficient) is -1 and when the a value is negative, then the parabola faces downward.
The axis of symmetry is x=4 and I know because this equation is in vertex form so I need to know what number will add into the bracket (x-4) to isolate x and the number is 4.
The optimal value is 12 and I know because in vertex form (y=a(x-h)^2+k) the k value is the optimal value so that is why the optimal value is y=12
Now that I know the axis of symmetry and the optimal value, I know the vertex since the axis of symmetry is the x value of the vertex and the optimal value is the y value of the vertex so the vertex is (4,12).
I also know that the vertex is the maximum ( because the parabola is facing downward).
Possible x-values are a set of all real numbers. Possible y-values are all real numbers less than 12. ( since the optimal value is 12, with the parabola opening downward).
Second Differences and Quadratic Relations
Now, I know that if the second differences are constant, then the graph represents a quadratic relation. To calculate the second differences, we must first calculate the first differences.
When you have table of values of x and y, you can determine whether or not the relationship is quadratic by using first and second differences.
Step 1: To find the first difference, subtract one y-coordinate from the one above it.** Remember to skip the first row since there is nothing above the first row. Continue this until the column is filled. If the first differences are the same, then the relation is linear.
Step 2: To find the second differences, subtract one first difference from the one above it.** Remember to skip the first 2 rows. Then, if the second differences are the same, that means that the relation is quadratic. If the second differences are not constant, then the graph represents a relation that is not quadratic or linear.
Lets do an example!
Parabolas go through transformations to become the 'final parabola' as a result. The three transformations parabolas go through are Stretching or Compressing, Reflecting, and Translating. The order that the transformations come in is Stretch first, then Reflect then Translate, and SRT is an easy way to remember this.
Stretch - The stretch is the first transformation parabolas go through before any of the others. The stretch refers to the a value in the standard, factored or vertex form equation. For a parabola to be stretched, the a value must be greater than 1. For example, the parabola y=3x^2 (a value is 3) is stretched vertically. For a parabola to be compressed, the a value must be less than 1, but greater than 0. For example, the parabola y=0.5x^2 (a value is 0.5) is compressed vertically.This includes decimals and fractions. In both cases, stretch and compression, the step pattern is altered. The basic step pattern is over 1, up 1 and over 2, up 4. With the stretch/ compression factor, the vertical (up/down)part of the step pattern is changed. The stretch factor is multiplied by the vertical part, and the outcome is the new step pattern.The step pattern helps lead to points that lie on the parabola.
For example a quadratic equation is y=3(x-4)^2+4. since the a value is three, the step pattern changes so I will multiply the a value to the vertical part of the step pattern (up/down).
Over 1, up 1x3= 3
Over 2, up 4x3= 12
Therefore, in this case, the step pattern is over 1, up 3 and over 2, up 12.
Reflect - In grade 10, the only reflection learned was a reflection occurring in the x-axis. When a parabola is reflected, the direction of opening is flipped. For example, a parabola can start off with facing upward, but if the parabola is reflected in the x-axis, the result is the parabola facing downward. A hint to know if a reflection occurs to see if the a values sign (-/+) is opposite in the starting equation and the final equation for the parabola.
For example, the equation y=x^2 transforms to the equation y=-x^2. With this, the starting equations a value was positive and the final equations a value is negative. Therefore, a reflection in the x-axis occurred.
Translate - This is the last transformation that is applied to the parabola. A parabola can translate horizontally and vertically. A vertical translation moves the parabola left and right and the horizontal translation moves the parabola up and down. In the equation y=a(x-h)squared+k, h is the variable that represents the horizontal translation and k is the variable that represents the vertical translation. If k is positive, then the parabola is translated up, but if k is negative, then the parabola is translated down. If h is negative, then the parabola moves to the right, and if h is positive, the parabola moves to the left. This is because when you try and isolate x in the bracket, h must be removed and to do so, you must add h if h is negative, and subtract h if h is positive.
For example, the quadratic equation transforms to the quadratic equation y=(x-4)^2+4, the bracket is (x-4)
Now, to isolate the x, I will have to get rid of -4 by adding 4.
since +4 is the number I add to the equation, that means that the parabola is going to translate 4 units to the right.
And the k value is +4, so the parabola is translated 4 units up
Therefore, this parabola is translated 4 units to the right and translated 4 units up.
1) the a value is 4 which means the graph stretches by a factor of 4.
2) the a value is negative so it reflects in the x-axis
3)the h value is -6, so the graph shifts 6 units to the right
4)the k value is -3, so the graph shifts down 3 units
Types of Quadratic Equations
Every type of quadratic expression/equation has a y-intercept because if the arms of the parabola are extended, the parabola will eventually hit the y- axis. To find out the y-intercept in each form, you must set x=0 and then continue to solve.
This section shows you how to find the vertex and x-intercepts of parabolas in each form.
1. Vertex form: The basic quadratic equation in vertex form is; y= a(x-h)²+k
Remember that the Axis of Symmetry in vertex form can be found by simply looking at the equation. The axis of symmetry is the x value in the vertex so if the equation is y=(x-2)²+3, then we need to find the vertex. I know that to get rid of the -2 in the bracket, I need to add 2. This 2 is the x-value of the vertex and the y value=3.Since the vertex is (2,3) then the axis of symmetry would be 2. Th optimal value of an equation in vertex form can also be found by looking at the value of k. In the equation y=(x-2)²+3, the k value is 3 and therefore, the optimal value is y=3. Recall that the optimal value is also the y-value in the vertex.
To find the x-intercepts in vertex form, the method called isolating for x must be used. Look at the "isolating for x" part in the vertex form section to understand the method more and to see an example.
3. Factored form: The basic quadratic equation in factored form is; y=a(x-r)(x-s)
In this form, the a value represents the stretch value. If 'a' is negative, then that indicates that the graph is facing downward. If 'a' is positive, then the graph will be facing upward.
To find the x-intercepts, look at both brackets.
Example: y= (x-4)(x-2) Each bracket represents an x-intercept so if one bracket is (x-4), then you must figure out which number should be added to isolate x and in this case, 4 is the number that will do so. So, one x intercept is (4,0).
The other bracket is (x-2):
To find the vertex of a parabola using factored form, you must first find out the axis of symmetry and the optimal value. To find the axis of symmetry, you must add both r and s together, then divide the sum by two. So, axis of symmetry: x= r+s/2
Once you have obtained the axis of symmetry, you must find the optimal value by substituting the axis of symmetry into the equation.
For example, if my axis of symmetry was 3 and my equation was y= (x-4)(x-2), then I would substitute 3=x into the equation and solve for y:
Therefore, in this case, the vertex is (3,-1) and the x-intercepts are (4,0) and (2,0).
Finding an equation when given the vertex
Sometimes, a question will give you the vertex with another point on the parabola and will ask you to figure out the equation. Here is an example:
Example #1: The vertex of a parabola is (-4,3). The parabola goes through the point (-5.1).
In the first example, the a value is 1, which means that it can follow the basic step pattern.
In the second example, there is a negative a value in the equation and the a value is -2, which means it cannot follow the basic step pattern.
A value represents the stretch value. If the a value is negative, the parabola will open downward and if the a value is positive, the parabola will open upward.
K represents the vertical translation and h represents the horizontal translation
Following the translations of h and k will lead one to the vertex of the parabola. Once the vertex is plotted, the step pattern can be followed. Therefore, vertex form gives the vertex, axis of symmetry, and optimal value of the parabola.
A zero of a parabola is another name for the x- intercepts and in order to find the x- intercepts you must set y=0. Please look at the types of equations to understand how to find the x-intercepts of a quadratic relationship in factored form.
When the (a) value changes the zeroes do not change.
When the (a) value changes the axis of symmetry does not change
When the (a) value changes the optimal value changes as well.
The first thing to graph with standard form is the y-intercept. In the equation, the c value is the y-intercept.
(Side note: There are two videos for example one so watch both parts to understand the full solutions)
(Side note: There are two videos for example one so watch both parts to understand the full solutions)
Graphing and Factoring relate because when we are trying to graph an equation in standard form, we must factor or use the quadratic formula to find the x-intercepts that we need to plot on the graph. Factoring is a method used to go from standard form to factored form and once we have factored form, we can set each factor to 0 so that we can find the x-intercepts. We can also just sub in the values of a, b, and c into the quadratic formula to get the x-intercepts of an equation in standard form. Without factoring (or using quadratic formula) a standard form equation, we wouldn't be able to plot the zeroes on our graph.
Factoring and the Quadratic Formula both relate to one another because both are used to find the x-intercepts of an equation. Factoring is used to go from standard form to factored form. When we have factored form, we can find the x-intercepts of a parabola by setting each factor to 0. With the quadratic formula, we are using an equation from standard form and substituting the values of a, b, and c to find the x-intercepts. Both methods get me the x-intercepts of an equation. In one of my videos, I use both methods to find the x-intercepts so go watch the videos to see and example.
The discriminant tells us how much solutions an equation (in standard form) will have. We learned that when D=0, there is only one solution. If D>0, then there is 2 solutions. If D<0, then there are no solutions. Solutions is another word for x-intercepts. When a parabola meets with the x-axis, the point of the coordinate at which they meet is the x-intercept. With this, we can easily know how much zeroes a parabola has before doing any other calculations. For example, if D=0, then I would right away know that there is only one x-intercept.
There are three forms of quadratic equations:Factored form,vertex form and standard form. All of them have the a value in common. When we want to go from standard form to vertex form, we use completing the square. Once we are done completing the square, we have 2 equations that represent the same parabola. And in these 2 equations, the a value (leading coefficient) will be the same no matter what.
For example, in the completing the square section, I had the equation Y=3x^2+36x-3 (standard form). When I completed the square, I got Y=3(x+3)^2-30 (vertex form).And this equation in factored form is y=3(x-0.08)(x+12.08).The similarity between these three is the a value. The a value never changed and was constant in all three equations. This means that no matter what form the equation is in, we will know if the parabola is being stretched or compressed based on the a value.
Completing the Square and Perfect Square Factoring relate because when we are completing the square, we use perfect square factoring. When we want to go from standard form to vertex form, we use completing the square. One step in completing the square is to add and subtract a certain number into the bracket (review completing the square section). When we add and subtract that number, we then take out the subtracted number out of the bracket by multiplying it with the a-value. Then, we are left with a bracket containing a perfect square trinomial. We use our knowledge on perfect square factoring in completing the square
These two topics relate to each other because when we use completing the square to go from standard form to factored form, we are finding out more about the characteristics of the parabola. For example, before completing the square, my equation was Y=x^2+4x+5. With this equation, I only knew the y-intercept was (0, 5) because of the c value. With just a first glance, that is all I know. After completing the square, my equation was Y=(x+2)^2+1(look at completing the square section to see how to complete the square). When I see this equation, I can tell right off the bat that the vertex is (-2, 1). The axis of symmetry is x=-2. The optimal value is y=1 and the optimal value is the minimum. The parabola faces upward (a value is +). Possible x-values are all real numbers. Possible y-values are a set of real numbers greater than 1. I found out all of this by looking at the equation.
Graphing and Equations relate because standard form, factored form, and vertex form are all able to be graphed.From each type of equation we need to find the x- intercepts then find the y and x value of the vertex. We use different methods for all three forms, but no matter what, we can graph all three forms. With vertex form, we know the vertex by looking at the equation, and we find out the x-intercepts by isolating for x. In factored form, we find out the zeroes by setting each factor equal to 0 and we can find the x value of the vertex (axis of symmetry) by adding the zeroes together and dividing by 2. We find the y value of the vertex ( opt. value) by subbing in the axis of symmetry from the step before. In standard form, we can factor/use quadratic formula to get the x-intercepts.We can find the x-value of the vertex by adding the zeroes together (the x-int's) and dividing by 2. Then, to find the y- value of the vertex, we sub in the axis of symmetry from the step before. These enable you to plot three points so you can graph.
I had the most trouble grasping the idea of Factoring. It was difficult for me to distinguish which type of factoring to use when I had to solve an equation.One thing I disliked about factoring was factoring complex trinomials. I did not like this part because we used the guess and check method, which was time consuming, and difficult to perform. It was not an accurate way, and took a lot of time because we had to guess the answer, instead of using a method that gave me the answer right away. Also, I always had a hard time identifying which factoring method to use. For example, in the test below, I got half a mark off of 7) because I didn't factor the difference of squares due to the fact that I did not notice that I could further factor after common factoring. This is why I did poorly on the "Mini Test 2:Quadratics"