Study Guide: Complex Numbers, Quadratic Equations, and Exponential & Logarithmic Functions

Complex Numbers and Their Operations

Definition and Arithmetic of Complex Numbers

Complex numbers extend the real number system by including the imaginary unit i, where $i$ is defined as the square root of -1. A complex number is written in the form $a + bi$, where $a$ and $b$ are real numbers.

• Addition/Subtraction: Combine like terms: $(a + bi) + (c + di) = (a + c) + (b + d)i$

• Multiplication: Use distributive property and $i^2 = -1$: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

• Division: Multiply numerator and denominator by the conjugate of the denominator: $\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)}$

• Example: $(2 + 3i) + (4 - 2i) = 6 + i$

Radicals with Negative Radicands

When simplifying radicals with negative radicands, express the result in terms of i.

• Key Point: $\sqrt{-a} = i\sqrt{a}$ for $a > 0$

• Example: $\sqrt{-16} = 4i$

Powers of i

The powers of i repeat in a cycle of four:

• $i^1 = i$

• $i^2 = -1$

• $i^3 = -i$

• $i^4 = 1$

• General Rule: $i^n$ depends on the remainder when $n$ is divided by 4.

• Example: $i^7 = i^{4 \times 1 + 3} = i^3 = -i$

Checking Solutions in Complex Numbers

To verify if a complex number is a solution to an equation, substitute it into the equation and simplify both sides.

• Example: Is $2 + i$ a solution to $z^2 - 4z + 5 = 0$? Substitute $z = 2 + i$ and check if the equation holds.

Quadratic Equations and Functions

Completing the Square

Completing the square is a method to rewrite a quadratic equation in the form $(x + p)^2 = q$.

• Steps:

  1. Move the constant term to the other side: $x^2 + bx = c$

  2. Add $(\frac{b}{2})^2$ to both sides: $x^2 + bx + (\frac{b}{2})^2 = c + (\frac{b}{2})^2$

  3. Rewrite as a square: $(x + \frac{b}{2})^2 = c + (\frac{b}{2})^2$

• Example: Solve $x^2 + 6x + 5 = 0$ by completing the square.

Solving Quadratic Equations

Quadratic equations have the form $ax^2 + bx + c = 0$. They can be solved by factoring, completing the square, or using the quadratic formula.

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Example: Solve $2x^2 - 4x - 6 = 0$ using the quadratic formula.

Applications of Quadratic Functions

Quadratic functions model various real-world scenarios, such as projectile motion and area problems.

• Example: The height $h$ of a ball thrown upward can be modeled by $h(t) = -16t^2 + vt + s$.

Functions: Composition and Inverses

Composite Functions

The composition of two functions $f$ and $g$ is written as $(f \circ g)(x) = f(g(x))$.

• Key Point: Always apply the inner function first.

• Example: If $f(x) = 2x + 1$ and $g(x) = x^2$, then $(f \circ g)(x) = f(g(x)) = 2x^2 + 1$.

Evaluating Composite Functions

To evaluate a composite function at a specific value, substitute the value into the inner function, then use the result in the outer function.

• Example: Using the previous functions, $(f \circ g)(3) = f(g(3)) = f(9) = 19$.

Inverse Functions

An inverse function "undoes" the action of the original function. If $f$ and $g$ are inverses, then $f(g(x)) = x$ and $g(f(x)) = x$.

• Finding the Inverse: Replace $f(x)$ with $y$, solve for $x$ in terms of $y$, then swap $x$ and $y$.

• Example: If $f(x) = 3x + 2$, then $y = 3x + 2 \Rightarrow x = \frac{y - 2}{3}$, so $f^{-1}(x) = \frac{x - 2}{3}$.

Determining if Two Functions are Inverses

To check if $f$ and $g$ are inverses, verify that $f(g(x)) = x$ and $g(f(x)) = x$ for all $x$ in the domains of $f$ and $g$.

• Example: If $f(x) = 2x - 1$ and $g(x) = \frac{x + 1}{2}$, then $f(g(x)) = 2 \cdot \frac{x + 1}{2} - 1 = x + 1 - 1 = x$.

Exponential and Logarithmic Functions

Translations of Exponential Functions

Exponential functions have the form $f(x) = ab^{x - h} + k$, where $h$ and $k$ translate the graph horizontally and vertically.

• Horizontal Translation: $h$ shifts the graph right (if $h > 0$) or left (if $h < 0$).

• Vertical Translation: $k$ shifts the graph up (if $k > 0$) or down (if $k < 0$).

• Example: $f(x) = 2^{x - 3} + 4$ is shifted 3 units right and 4 units up compared to $f(x) = 2^x$.

Applications of Exponential Functions

Exponential functions model growth and decay, such as population growth or radioactive decay.

• General Form: $y = ab^{x}$, where $a$ is the initial value and $b$ is the growth/decay factor.

• Example: If a population doubles every year, $P(t) = P_0 \cdot 2^t$.

Function Type	General Form	Key Feature
Exponential Growth	$y = ab^{x}$, $b > 1$	Increases rapidly
Exponential Decay	$y = ab^{x}$, $0 < b < 1$	Decreases rapidly

Additional info: Some topics, such as logarithmic functions, are implied by the chapter titles but not explicitly listed in the study guide. The notes above focus on the explicitly mentioned topics.