There are three ways to solve and write the quadratic equation.

General Form

The general form is y =ax² + bx + c

This form tells us the y-intercept, which is c. It tells us the scale, which is a, that is what number changes the 1, 3, 5 scale. This equation does not help us graph it, it tells us it is quadratic by the x², but we can only graph it if it is in factored or standard form.

y =x² + 6x + 9 is an example of a quadratic formula in general form.

Factored Form

Factored form is the first form that is able to be used to solve quadratic equations and graph them. When an equation is formed to factored, it tells you the x-intercepts. It also tells you the scale.

y =x² + 6x + 9 can be factored to y = (x+3)(x+3), which means the only x-intercept is -3, found by taking the factors and solving them to 0 (x+3 = 0, x + -3). This also tells us this is up facing and has the scale 1, 3, 5. Since this quadratic crosses x only once, we can assume it starts there and goes up.

y =x² + 7x + 10 can be factored to y = (x +2)(x+5), so the x-intercepts are -2 and -5 ( x+2 = 0, x+5 =0). This doesn’t help us too much to fully graph because we know where the x-intercepts are, it is up facing, and has the general scale, but we do not know where it starts.

Standard or Vertex Form

This is the most helpful form for graphing, y = (x – p)² + q, (p,q) is the vertex (x,y).

If we have an equation in general form, we can change it to standard form by finding the square.

y =x² + 6x + 8

Find the square means taking the middle term, dividing it by two and then taking it to the power of 2 and then adding that as a 1 (a positive form and negative form). This helps create the factor.

y =x² + 6x + 8

6/2 = 3

3² = 9

A 1 is like 1/1 just like 3/3 is also just 1.

We can then add the 9 back in as a 1, so +9 and -9.

y =x² + 6x +9 -9 + 8

We can then create x² + 6x +9 as a factor (x+3)²

y =(x+3)² -9 + 8

y =(x+3)² -1

We have now a new form that can help us factor. This form tells us the vertex which is (p,q). We also must remember that since the equation has a negative p, the term in its place has been multiplied by a negative. So, when looking for the vertex, it would be (-3, -1). We then find this point of the graph and start from there going up as indicated since there is no negative before the bracket affecting the scale, and going up by 1, 3, 5.

Quadratic Formula

The last way to solve a quadratic equation that cannot be factored is using the quadratic formula. 

y =x² + 3x – 4

First, we list a = 1, b = 3, c = -4

Then it is just filling in the equation.

x = -(3)±√(3)²−4(1)(-4)/2(1)

x = -3±√9+16/2

x = -3±√25/2

x = -3+5/2 0r x = -3-5/2

x = 1 0r x = -4

The parent function of a parabola is y = x²

Vertical translation is shown when there is a term added to x². So, if y = x² + 5, the parent function would be moved 5 units up.

If y = x² – 7, the parent function would be moved down 7 units.

Horizontal translations are shown when y = (x – p)², p being the translation. If the parent function moves 5 to the right, we replace p with 5, y = (x – (5))²

If p = -4, we replace p and the graph moves 4 to the left, y = (x – (-4))² or y = (x +4)²

Now, if we add this together, we can understand how a parent function changes.

y = (x +4)² – 5

This means that the horizontal translation is 4 units to the left, then 5 units down.

The scale can affect whether the parabola is stretched or compressed. If there are no terms before the brackets, the scale is as original, 1, 3, 5, etc. If there is a number higher than 0 in front of the brackets, y = 2(x +4)² – 5, the parabola becomes stretched.

If the number is less than 1 and greater than 0, y = .2(x +4)² – 5, the parabola is compressed.

Lastly, if this term before the brackets is negative, the parabola becomes down facing, y = -(x +4)² – 5

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