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Standard Form
The standard form equation of a parabola is y=ax^2+bx+c(look back for reference if needed). We will cover how to solve the zeroes, axis of symmetry and optimal value of a parabola with this equation. You will also learn how to convert this equation into vertex form and factored form.
Converting into Factored Form
To convert a standard form equation in to factored form you will need to follow these steps:
We first need an example equation, we will use y=x^2+6x+8, I will break it down in to multiple steps...
Step 1- Put these values in a chart, like this for example...
6x
x^2 8
Step 2- You need to find 2 numbers that multiply to "c", but add to "b" at the same time, in our case we need to find 2 numbers that multiply to 8 but add to 6. 4 and 2 work, so now we do this...
6x
x^2 8
4
2
x
x
Step 3- I put 2 x's in the x^2 section of the chart because 2 x's multiply to x^2 incase you were wondering why I did that. Now we have the standard form equation in factored form, all you do now is write it like this...
y=(x+4)(x+2)
"Now this is great and all but what do you do if there were negative values in the standard form equation? Or what if there is an a value in the equation instead of it just being a 1?" Well it does not change much, so if I have this equation, i.e -x^2+6x-8, you solve it like this...
Step 1- You cannot have a negative leader(leader means x^2), so what you do is divide the whole equation by a negative one(as there is no actual integer after the negative sign), so...
y=-x^2/-1+6x/-1+8/-1
y=x^2-6x+8
Step 2- Into the chart...
-6x
x^2 8
Step 2- Now solve...
-6x
x^2 8
-4
-2
x
x
Step 3- Write it out...
y=-(x-4)(x-2)
Since we originally divided the whole standard form equation by a(-1), we multiply our final answer by a as well. -4 and -2 add up to -6, and at the same time they multiply to 8, so this works. Now if we have an equation like this, i.e 2x^2+6x-8 things change a little bit.
Step 1- Firstly, always, always, always look whether the b and c values can be divided by the a value, in our case they can, so...
y=2x^2/2+6x/2-8/2
y=x^2+3x-4
Step 2- Into the chart...
3x
x^2 -4
Step 3- Solve...
3x
x^2 -4
4
-1
x
x
Step 4- Write it out...
y=2(x+4)(x-1)
Since we originally divided the whole standard form equation by a(2), we multiply our final answer by a as well. Now if we were given a standard form equation where b and c cannot be divided by a, i.e 3x^2+8x+4, it gets a little more complex.
Step 1- Into the chart...
8x
3x^2 4
Step 2- Find 2 numbers that multiply to 3, and 2 that multiply to 4, so...
8x
3x^2 4
3x
x
2
2
Step 3- Write it out...
y=(3x+2)(x+2)
You are probably asking how does 2+2=8? It doesn't we cross multiply in the chart, i.e 3x2=6 and 1x2=2, and at the same time, 2x2 still equals 4. The reason this is more complex is because sometimes certain values will not work and you have to find different ones, until you finally get the answer and it works out.
If you have tried everything and you still cannot get the standard form equation into factored form, it means it is impossible to do so then, i.e x^2+6x+120, that is impossible to convert into factored form. The process we used to convert standard form into factored form is called "factoring".
Cross multiply...
3x2=6
1x2=2
6+2=8 ✔
2x2=4 ✔
∴ the factored form equation is, "y=(3x+2)(x+2)".
Converting into Vertex Form
Axis of Symmetry and Optimal Value
To convert a standard form equation into vertex form we will need to "complete the square", I will teach you what that is by seperating this into steps.
We will use the equation -2x^2+8x+16, we're using this equation specifically because it deals with a negative integer in front of x^2, this will be the most difficult type of question you will face when converting standard form into vertex form, so if you can solve this you can solve all variations of this question.
Step 1- Put "ax^2+bx" in brackets...
y=(-2x^2+8x)+16
Step 2- Divide the values in the bracket by "a"...
y=(-2x^2/-2+8x/-2)+16
y=-2(x^2-4x)+16
Since we divided the values in the brackets by a, a will come outside of the bracket
Step 3- Divide "b" by 2 then square it...
-4/2=-2
-2^2=-4
Step 4- Put the previous value you solved into the brackets, put a positive and negative version of your previous answer(completing the square) like this...
y=-2(x^2-4x+4-4)+16
Step 5- You take the negative out and put it next to c, if there is an a value, you multiply it by the negative number, in our case there is so we do this...
-2(-4)=8
y=-2(x^2-4x+4)+16+8
Step 6- Simplify the equation further by adding the numbers outside of the bracket...
y=-2(x^2-4x+4)+24
Step 7- Factor the equation inside of the bracket(if you do not know what that is read the previous lesson on converting standard form into factored form)
-4x
x^2 4
To solve for the axis of symmetry(x) we will use the equation "x=-b/2a", with the standard form equation we already know the a and b values, so you just substitute them into this equation so...
Example equation: y=2x^2+4x+8
x=-b/2a
x=-(4)/2(2)
x=-4/4
x=-1
Now we have our axis of symmetry, to solve for the optimal value(y) we substitute x into the standard form equation, so...
y=2(-1)^2+4(-1)+8
y=2(1)-4+8
y=2-4+8
y=-2+8
y=6
∴ x=-1 and y=6. Using these 2 values we now know our vertex for this standard form equation which is (-1, 6), that is how you solve for the axis of symmetry and optimal value when given the standard form equation of a parabola.
Zeroes(x-intercepts)
To solve for the x-intercepts with the standard form equation requires you to use an equation called "The Quadratic Formula", the equation itself is "x=-(b)±√b^2-4(a)(c)", we have to substitute the values from the standard form equation into this one.
2(a)
I will do one example and break it down into different steps, using the equation "y=2x^2+8x+6"
Step 1- Substitute the values into the quadratic formula...
x=-(8)±√8^2-4(2)(6)
2(2)
Step 2- Solve the "b^2", in our case 8^2, so...
x=-(8)±√64-4(2)(6)
Step 3- Do all the multiplications...
x=-8±√64-48
2(2)
4
Step 4- Add the 2 values in the square root...
x=-8±√16
4
Step 5- Solve the square root...
√16=4
x=-8±4
4
Step 6- Plus and minus the -b by the square rooted answer...
x=-8+4
x=-4
x=-8-4
x=-12
Step 7- Divide both those values by the 2(a) value...
x=-4/4
x=-1
x=-12/4
x=-3
∴ the x-intercepts of y=2x^2+8x+6 are -1 and -3. Remember that the b^2 value in the quadratic formula cannot be negative, even if b is negative in the standard form equation it is still positive in the square root, but it is only positive in the square root, if it is negative it is still negative everywhere else in the quadratic formula but in the square root.
Discriminant
The "discriminant" decides how many x-intercepts there will be in a standard form equation, it is the square root value in the quadratic formula. The equation for the discriminant is "D=b^2-4(a)(c)".
If D is > than 0, there are 2 x-intercepts.
If D is < than 0, there are 0 x-intercepts.
If D=0, there is 1 x-intercept.
-2
-2
x
x
Step 8- Write it in vertex form...
y=-2(x-2)^2+24
The reason "(x-2)" is squared is because there are 2 (x-2)'s in the chart. Now you are done.
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