Key Equations
What is the formula for the standard form of a quadratic?
y=ax^2+bx+c
Find the vertex for the following quadratic
y=-(x+4)^2-4
(-4,-4)
What is the procedure we use to convert a quadratic from vertex form to standard form?
Completing the square
What comes once in a minute, twice in a moment, but never in a thousand years?
The letter "M".
What is the formula for the vertex form of a quadratic?
y=a(x-h)^2+k
What is the y intercept of the quadratic?
(0,-20) Could you sketch the quadratic with this information? (The vertex and Y-intercept) (hint, hint)
What does
a
tell us about the quadratic?
If a is negative the parabola bends down (like a Sad face). If a is positive the parabola bends up (like a U).
What was the name of the mathematics "program" we watched a video on after the first exam?
The Langlands Program
What is the quadratic formula?
x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
Convert the quadratic from above to standard form. What do you obtain?
y=-x^2-8x-20
What is the point, like (x,y), that tells us the vertex from the formula for a quadratic in vertex form?
(h,k)
Why are functions important?
They help us model real world scenarios and predict what will happen next in those scenarios.
Compute
\sqrt{-8}
\pm2i\sqrt{2}
What is the discriminant of the quadratic from above?
\Delta=\sqrt{-16}
How can you determine if a quadratic has real or imaginary roots by looking at its graph?
If the quadratic does not intersect the x-axis, it has imaginary roots
Who was the first mathematician to come up with rules for multiplying imaginary, complex, numbers?
Bombelli in 1572
What is the formula for the discriminant of a quadratic?
\Delta=\sqrt{b^2-4ac
Find the roots of the quadratic from above. What are they?
i\sqrt{2}-4=x
-i\sqrt{2}-4=x
What do the different values of the discriminant tell you about the solutions of a quadratic?
If the discriminant is negative, the quadratic has no real solutions. If it is zero, there is one solution, x=\frac{-\b}{2a , If it is greater than zero there are two solutions,
x=\frac{b-\sqrt{\Delta}}{2a} and x=\frac{-b-\sqrt{\Delta}}{2a}
Who was the first mathematician to realize that the solution to a quadratic equation yields two roots?
Brahmagupta, the Indian mathematician in 700 A.D.