Quadratics

Tavleen Mann

Background Check! What are Quadratics?

The word quadratic comes from a latin word, quadratus, which means square.

A quadratic relationship is always symmetrical. Same on the left, and same on the right. The graph for a quadratic relation has a parabola. The parabola can either open up, creating a smiley face if it is positive, or open down, creating a sad face if it is negative.

Terms used in Quadratics

Vertex: The maximum or minimum point on the graph. It i the point where the graph changes direction. Labelled as (x,y).

Minimum/Maximum Value or Optimal Value: The highest or lowest point on the parabola, the y-value on the vertex. Labelled as (y=#).

Axis of Symmetry: A line that initially "cuts" your parabola in half, the x-value on the vertex. Labelled as (x=#).

Y-Intercept: Where the parabola meets the y-axis. Labelled as (0,#).

X-Intercept(s): Where the parabola meets the x-axis. There can be 0,1 or 2 intercepts. Also called "zeros". Labelled as (#,0).

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In order for a relation to be quadratic, they need to have the same second differences.

If all the first differences are the same, then the equation is not quadratic, it is linear. And if the second differences are not the same either, the equation is not quadratic.
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Vertex Form

Step Pattern

The original/base step pattern of a parabola is "over 1 up 1, over 2 up 4" and so on. The s value is generally squared to get the y value. The vertex tell us where the step pattern starts and the "a" value decides if the step pattern is vertically stretched or compressed. The step pattern allows us to plot the point, and make our parabola by connecting the points based on the step pattern itself.
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This parabola follows the original step pattern of over 1 up 1, with the vertex of (0,0) graphed using desmos.

Transformations

y=a(x+h)-k

If the "a" value is more than 1, the parabola is stretched, and if the "a" value is less than 1, then the parabola is compressed. A parabola can also be reflecting the x-axis. This simply means that the parabola is opening downwards and is negative. The only difference in the step pattern would be that the y-value would become negative. The "h" value is the number that is after "x" in the equation. It makes the parabola have a horizontal shift (left or right). A negative "h" value shifts right and a positive "h" value shifts to the left. The "k" value is the number after the brackets containing the values of "x" and "h". The "k" value has a vertical shift (up or down). A negative "k" value shift down whereas a positive value shift up.
For example, with this equation, we have a vertex of (4,6), since the horizontal shift is right 4 units and the vertical shift is up 6 units. The step pattern will the same as the original as there is no given "a" value in this equation.
graphing factored form

vertex equation

how to find a

Factored Form

In order to graph factored form, you need to find the x-intercepts, and the vertex, which includes the optimal value and the axis of symmetry. To find the x-intercepts, we need to set the y value to 0. We then isolate the variable to find its value. After finding the 2 x-intercepts, we can find the axis of symmetry by adding the 2 values together and dividing the sum by 2. The optimal value of the parabola is the max or min value being represented, which is the lowest or highest point shown. The optimal value is the k value shown in an equation. It is also the y intercept shown in a point. To find the optimal value, you have to substitute the axis of symmetry for the x value in the equation, to solve for y.

Equation:

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Multiplying Binomials

To multiply binomials, we can use a visual interpretation of tiles as shown below. Mathematically, we can expand the 2 brackets which usually gives us an answer of a trinomial.
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Common Factoring

To make it easier to come up with a solution for an equation, we can common factor out the GCF of an expression.

Step 1: Find a GCF that all terms have in common

Step 2: Divide the expression by the GCF which means the number will now be placed outside of the brackets.

Step 3: Simplify and leave the GCF outside of the bracket for your final answer.

Common Factoring

Factoring Trinomials by Grouping

Setting 2 sets of brackets by grouping
Factoring by Grouping

Factoring Simple Trinomials

x²+bx+c

In order to change a simple trinomial into factored form, we need to:

Step 1: Make sure that the format of the equation is in the equation written above. Ex. x²+5x+6.

Step 2: Find 2 factors that multiply to give the sum of c. Ex. 2 and 3 multiply to give us 6, which is the c value, which means these factors work.

Step 3: Make sure the factors add to equal the b value. Ex. 2 and 3 add to give us 5, which is the b value, so these factors work.

Step 4: Write the equation out in factor form with the factors put into the brackets, in a order that gives us the answer of the simple trinomial we first started with. Ex. (w+2)(w+3) works since it gives us the original simple trinomial we started with.

Simple Trinomial Factoring

Factoring Complex Trinomials

Complex trinomials will have an a value in front of the x², which means that when factoring, we follow the same steps as factoring simple trinomials, with the difference of adding a number in front of the x value after the trinomial is factored. The factors added in front should still make sense and when solved, should give us the original answer of the trinomial we began with.

Ex: 8x²+4x-5 becomes (4x+5)(2x-1).

Complex Trinomials

Perfect Squares

A perfect square trinomial always starts and ends with a square and looks like, (a+b)²= a²+2ab+b²
Example 1:
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Example 2:
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The equation cannot be a perfect square, if one of the variables can't be perfectly squared
Perfect Squares

Difference of Squares

Difference of squares is similar to perfect squares, but the difference is that the signs in the two sets of brackets are opposite from each other. The trinomial looks like a² - b²= (a+b) (a-b)
Difference of Squares

Word Problems

The following word problems have different ways on how to solve them. The word problems are used for different scenarios. Some follow certain equations, while others follow their own. Multiple word problems are shown below with thorough answers.
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Standard Form

  • A way to write a quadratic relation
  • y=ax²+bx+c
  • An example in standard form is y=3x²+15x+18

The Quadratic formula is...

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How to find the zeros using the quadratic formula

The square root should be written as both positive or negative since we use both to get 2 different answers in the end. Both zeroes will be given, but it is up to us to choose which one fits the problem better.
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Discriminants

  • b²-4ac
  • A discriminant is the equation inside of the square root
  • The discriminant tells us how many solutions there are, if there are any
  • If the discriminant is negative, then we know there will be no solutions as a negative number cannot be square rooted
  • If the discriminant is 0, then there is only 1 solution, since a 0 would not change if you either add or subtract
  • If the discriminant is greater than 0, there will be 2 solutions, like the example above

Axis of Symmetry

  • To find the axis of symmetry, we use the equation outside of the discriminant in the quadratic equation
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Optimal Value

  • Substituting the axis of symmetry found into the original equation, we get y, which is the optimal value.


Refer to the example above, for the answer below

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Completing the Square

  • Completing the square means converting from Standard form to Vertex form
  • y=ax²+bx+c to y=a(x-h)²+k
Completing the Square - Solving Quadratic Equations

Reflection

I've learned many new things in the quadratic unit. Many of the things I've learned, have had multiple connections to the real world as well. I felt that the mini unit 1 and 3 were a little easier to understand than mini unit 2. This unit was very confusing for me to understand since there were many things I had to remember and understand. 1 thing that I found very difficult was perfect squares. I did not understand them at first, but with practice and a lot of help, I understood them. Something I found really easy from the start was common factoring. I understood it really well and did well with it.
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Using this assessment as an example, I find it easy to common factor an equation. Although there were many things I started off with roughly, I now have a better understanding of how quadratics work, and how they help in real life.