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Foundations of Quadratics
Vertex, standard, and factored form, axis of symmetry, concavity, quadratic formula
Vertex, standard, and factored form, axis of symmetry, concavity, quadratic formula
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The graph of a quadratic function has the general shape of a parabola.
It is symmetrical about the axis of symmetry and has a maxima or minima at the vertex, which lies on the axis of symmetry.
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A quadratic in x is an expression of the form
where a=0.
The concavity of a parabola describes whether it "opens" up or down.
The parabola corresponding to ax2+bx+c is:
Concave up if a>0
Concave down if a<0.
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Most quadratics can be factored as a product of linear terms:
We call the generalized form above factored form. Notice that α and β are the roots of the quadratic, since when x=α or x=β the expression will evaluate to zero.
We can factor quadratics in the form ax2+bx+c by splitting b into a sum α+β such that that αβ multiplies to ac.
After rewriting the expression as ax2+αx+βx+c, factor the first pair and the second pair separately. The common factor will emerge, and you can pull it out. The result is the fully-factored expression.
For example, in the quadratic 3x2+8x−3, we want to split 8 into α+β such that αβ=−9. We can do this by choosing α=9 and β=−1:
The roots of a quadratic correspond to the x-intercepts of its graph. When x=a or x=β, the entire expression equals zero, which is reflected on the graph.
The equation of the parabola below is −(x−α)(x−β):
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A quadratic in the form ax2+bx+c can be written in the form
for some h,k∈R.
For the quadratic ax2+bx+c, the parabola has an axis of symmetry at
The axis of symmetry is the vertical line dividing the parabola perfectly in 2. The x-coordinate of the vertex, h, is equal to the x value where the axis of symmetry is located.
To convert from the form ax2+bx+c to a(x−h)2+k:
The values for a will match up directly.
Use the axis of symmetry x=−2ab to find h=−2ab.
Equate c=ah2+k, and substitute the h found to find k.
For any quadratic ax2+bx+c, the roots of the quadratic can be found using the quadratic formula:
When the values of a,b or c are large enough that using the quadratic formula becomes difficult, a calculator can be used to find the roots.
Your calculator should have an app for solving quadratics.