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Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
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.
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.
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):
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.
Plug h into ax2+bx+c 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.
The discriminant of a quadratic is the term under the square root in the quadratic formula:
When Δ<0, the square root has a negative value inside, and so the quadratic has no real solutions.
When Δ=0, the square root is zero, and the ±√Δ makes no difference, so there is only one real solution.
When Δ>0, √Δ is positive and so ±√Δ yields two real roots.
A quadratic inequality is an inequality of the form
They can be solved by finding the roots of the quadratic and the concavity of the parabola.
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
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.
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.
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):
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.
Plug h into ax2+bx+c 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.
The discriminant of a quadratic is the term under the square root in the quadratic formula:
When Δ<0, the square root has a negative value inside, and so the quadratic has no real solutions.
When Δ=0, the square root is zero, and the ±√Δ makes no difference, so there is only one real solution.
When Δ>0, √Δ is positive and so ±√Δ yields two real roots.
A quadratic inequality is an inequality of the form
They can be solved by finding the roots of the quadratic and the concavity of the parabola.