The process of converting quadratic equations between different forms, such as standard, factored, and vertex forms, involves specific mathematical transformations. For instance, expanding and simplifying factored forms leads to the standard form, while determining the axis of symmetry (
Standard to vertex.
x^2+15x+100
(x+20)(x-5)
AOS:-20+5/2=-15/2=-7.5
(-7.5+20)(-7.5-5)=
(12.5)(-12.5)=-156.25
(x+7.5)^2-156.25
Factored to vertex.
-2(x-5)(x+3)
AOS:5+(-3)/2=2/2=1
-2(1-5)(1+3)
-2(-4)(4)=32
-2(x-1)^2+32
standard to factored.
x^2+15x+100
(x+20)(x-5)
Vertex to standard.
2(x-4)^2-28
2(x-4)(x-4)-28
2(x^2-4x-4x+16)-28
2(x^2-8x+16)-28
2x^2-16x+32-28
2x^2-16x+4
factored to standard.
-2(x-5)(x+3),
-2(x^2+3x-5x-15),
-2x^2+4x+30,
Quadratics Assignment
Part 2
Vertex form characteristics
a(x-h)^2=k
h is the AOS. the h tells us how far left, or right, the slope has moved. k is the minimum or maximum value depending on if the parabola opens up or down. a indicates the optimum value. If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function, if the parabola opens down, the vertex represents the highest point on the graph, or the maximum value. To find the y-intercept, you simply plug in 0 as x in your equation and solve it. You create the equation by simply finding the AOS and the vertex on the graph and see if the graph opens up or down. Then look at the skeleton of the equation and put the numbers you find there. There are special cases like when there is no x-intercepts, the slope is above the x-axis line.
Part 1
vertex
From vertex to standard, expand your equation then add the terms you can together in the brackets. Then multiply a with the rest of the equation. Then add the terms without x together to get the standard form. From vertex to factored, convert your vertex to standard like show above, then convert from standard to factored as shown above too.
factored
From factored to standard, expand by multiplying both terms in the first bracket to the second bracket. Simplify the equation by adding the two terms that you can together Then you multiply A with the rest of the terms to find the standard form. From factored to vertex, find the AOS by adding your two numbers together and dividing it by 2 to get your first term. Use that number to replace x in the equation and solve it to get your second term. Once you have your vertex terms, put it in a vertex form equation.
standard
From standard to factored, you factor it by finding
two terms from c that equal b. From standard to vertex, you first factor it, then you find the vertex from the AOS by adding your two terms then divide it by 2. After, you plug the number that you got where the x is in the equation. Solve the equation normally to get your second number for your vertex. After you have the 2 numbers needed(h,k), put it where the numbers should be in a vertex equation.
