For starters...

Quadratics are used to find prices, trajectories and speed. The equations help us find out what the ending product might look like. They are also used to determine parabolas.

Graphing Vertex Form

Vertex form is referred to y=a(x-h)^2+k. The variable "a" is to determine if your graph will be stretched or compressed (e.g. if a<1 , then the graph will be compressed, but if a>1 then the graph will be stretched). The "h" is the horizontal shift on the y-axis of the graph. And finally, "k" is the vertical shift on the x-axis

To graph a vertex form, you will need to know the step pattern. The step pattern we use goes by 1,4,9,16. These are the number of steps we make when plotting points to construct the parabola. When plotting the points we always move 1 to the left from the vertex and 1 up, then move 1 left again, and this time 4 up, continuing the pattern. You also do this to the right side as well. To graph the equation, multiply the step pattern by "a", (e.g. if a=2, (1,4,9,16)x2).

To evaluate the parabola, we must look at certain things. We must look at the vertex, the axis of symmetry, the optimal value, and whether the parabola is concave up or down. To find the x and y values from the equation is simple. "h" is the x value, and "k" is the y value. They are written as (h,k) like (x,y). The axis of symmetry is only the "h". In the equation it is referred to "-h" so whatever the value as "h", the sign is always switched, (e.g. if y=-2(x-2)^2=4 then the axis of symmetry is x=2). The optimal value is "k" and is written as y=4. To determine if the graph is concave up or down, we look at "a".If "a" is positive then the graph will go up (concave up) but if "a"is negative then the graph will go down.

If we were to solve an equation, y=x^2+3. We can see that there is no number in front of "x" so we know that there is always 1. So your step pattern doesn't change. There also isn't an "h" so we use that as zero. The y axis is the only one changing. Then the "k" is 3, so we move 3 up from the vertex, since it's positive. Then you plot the step pattern points to finish the parabola.

Simple Trinomials-Factoring

To factor simple trinomials, we need to find common factors and then simplify the equation. For example, if we had x^2+5x-6 and we were told to factor, we need to find the 2 numbers that the middle term and the last term share. We want to find what multiplies to get -6 and what adds to get 5. So that would be 6, and -1 because 6x(-1)=-6 and 6+(-1) is 5. The factored form will look like (x+6)(x-1), because (x)(x) is x^2, (x)(-1) is -1x, (x)(6) is 6x, and (6)(-1) is -6, which equals to x^2+5x-6. Lets try another example.

Complex Trinomials-Factoring

This is the same concept, but with "a" or x^2 being more than 1. To factor the equation, you must find the common multiple. For example, if we had 5x^2+5x-30, we will need to find what's in common with every term. They all can be divided by 5. So the equation will look like 5(x^2+x-6) when divided. Note that the 5 is in front of the simplified equation. And from here you will simply factor it like above, keeping the 5. The final equation will look like 5(x-2) (x+3).

Perfect Squares-Factoring

This is when the equation can be square rooted. For example, if we had x^2-4x+4. They can be all square rooted. When following this procedure, we only square root the first and last term. We are left over with (x-2)^2. We put an exponent of 2, because this equation is written twice. We also get the sign from the previous equation.

Difference of Squares-Factoring

Difference of squares is when "a" minus a number. For example, x^2-4 is a difference of squares because there are only two terms and they have a subtraction symbol in between. To factor this, we must square root each term like we did in Perfect Squares, but we cannot square root a negative number so we represent them into two terms like factoring the Trinomials. We will get (x^2+2) representing the positive outcome, and (x-2) representing the negative outcome. Note that there is only one exponent in both brackets.

Solving Equations by Factoring

We can solve previous equations even more by graphing them. This will give us two points that we can plot on a graph to make a parabola. For example, if we had x^2+x-6 and we factored it, it will look like (x-2) and (x+3). We could make a parabola out of these equations and solve. All we need to do is take each set of brackets and set them to zero. x-2=0 and x+3=0. We then must solve for x by moving the numbers on the other side of the equal sign and changing the sign. For this equation we will be left with x=2 (2,0) and x=-3 (-3,0). We will be plotting this on the graph.

Next we can find the axis of symmetry by implementing a formula, -b/2a."b" sign must change when putting it in the formula. -1/2(1) is what we are left with. When you solve this, we will get -0.5. This will be our axis of symmetry. And finally, plot the point and draw a vertical line to indicate the axis of symmetry, and use the step pattern to complete the graph.

Quadratic Fomula

The Quadratic Formula is a formula derived from completing the square.
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This is the right side of the equation, b^2-4ac. Without the division.

b^2 - 4ac

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