Background Check! What are Quadratics?
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
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).
In order for a relation to be quadratic, they need to have the same second differences.
y=a(x+h)-kIf 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.
how to find a
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.
Factoring Trinomials by Grouping
Factoring Simple Trinomials
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.
Factoring Complex Trinomials
Ex: 8x²+4x-5 becomes (4x+5)(2x-1).
Difference of Squares
- A way to write a quadratic relation
- An example in standard form is y=3x²+15x+18
The Quadratic formula is...
How to find the zeros using the quadratic formula
- 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
- 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
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