Pre-Calculus Linear and Quadratic Functions and Modeling

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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|


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