Pre-Calculus Linear and Quadratic Functions and Modeling
What is the linear factored form of a function?
Use extracting roots, factoring, completing the square, or quadratic formula
When finding all zeros of a function using complex numbers...
Use the same zero again with the new function after finding a zero.
What is the range for a parabolic graph that opens down?
(-∞, k]
What is the sum of (a + bi) + (c + di)?
(a + c) + (b + d)i
What is the difference of (a + bi) - (c + di)?
(a - c) + (b - d)i (always use addition)
What is the vertex?
(h, k)
What is i²?
-1
What is i³?
-1
How do you find h/axis of symmetry?
-b/2a
What is the standard form of -√125?
-√(25 * 5) = -5√(5) + 0i
What is i⁴?
1
What is Factor Theorem?
A polynomial f(x) has a factor x - k if and only if f(k) = 0, e.g. x + 2 is a factor of 2x + 4, -2 is a zero of x + 2 and by using synthetic division, you find that it is a factor
What is the Fundamental Theorem of Algebra?
A polynomial function of degree n > 0 (cannot be less than 0) has n complex zeros.
What is a complex number?
Any number that can be written in the form a + bi (a is the real part and bi is the imaginary part)
How do you find the vertex of a function?
By finding the point where a function reaches a maximum or a minimum
What is Remainder Theorem?
If a polynomial f(x) is divided by x - k, then f(k) = r, e.g. f(x) = 2x³ - 4x² + 6; k = -3, f(-3) = -84 (using synthetic division)
If a and b are real numbers and b ≠ 0 and a + bi is a zero of f(x), then...
Its complex conjugate a - bi is also a zero of f(x)
How do you find the product of complex numbers?
Multiply the numerator and denominator by the denominator's conjugate.
Does form matter when finding the zeros/x-intercepts of a vertex?
No
What is the discriminant of a quadratic function?
The quantities within the square root of the quadratic formula.
If b² - 4ac > 0, then...
There are two distinct real solutions
If b²-4ac < 0, then...
There is a complex pair of solutions
If b² - 4ac = 0, then...
There is one repeated solution
How do you find the midpoint?
[Re(z₁) + Re(z₂)/2] + [Im(z₁) + Im(z₂)/2] Re: Real # Im: Imaginary #
What is the range for a parabolic graph that opens up?
[k, ∞)
What is the standard form of a complex number?
a + bi
What is an axis of symmetry?
an imaginary line that goes exactly in the middle of a parabola, x = h
What is an imaginary unit?
bi, where b is a real number and i = √-1
How do you find k?
f(-b/2a) or f(h)
What is the equation for vertex form?
f(x) = a(x - h)² + k
What are possible rational zeros?
factors of constant/factors of leading and last coefficients
What are the linear factors of (3 + 2i)?
x - (3 - 2i) and x - (3 + 2i)
What is the Quadratic Formula?
x = -b ± √(b² - 4ac)/2a
What is the complex conjugate of a complex number? z = a + bi
z = (a + bi)* or z = a - bi
How do you find the distance between z₁ and z₂?
|z₁ - z₂| or |z₂ - z₁|