Quadratic Equations

Leading Coefficients

The leading coefficient of a quadratic equation is the coefficient of the highest-degree term. For a quadratic equation in standard form, , the leading coefficient is a.

• Significance: The sign of the leading coefficient determines whether the parabola opens upward (a > 0) or downward (a < 0).

• Example: In , the leading coefficient is 2.

Vertex and Vertex Form of an Equation

The vertex of a parabola is its highest or lowest point, depending on the direction it opens. The vertex form of a quadratic equation is , where (h, k) is the vertex.

• Finding the Vertex: For , the vertex is at .

• Example: For , vertex at ; . So, vertex is (-2, -1).

Solving Quadratic Equations

Quadratic equations can be solved for real and complex solutions using several methods:

• Factoring: Express the quadratic as a product of two binomials and set each factor to zero.

• Completing the Square: Rewrite the equation in the form and solve for x.

• Quadratic Formula: For , solutions are given by:

• Example: Solve by factoring: so or .

Properties of Exponents

Exponents follow specific rules that simplify expressions involving powers.

• Product Rule:

• Quotient Rule:

• Power Rule:

• Zero Exponent: (for )

• Negative Exponent:

• Example:

Power Functions

Definition and Basic Properties

A power function is a function of the form , where a and n are constants.

• Domain and Range: Depend on the value of n (even, odd, positive, negative).

• Example: is a power function with domain and range .

Concavity and Monotonicity

• Concave Up: Graph opens upward (e.g., ).

• Concave Down: Graph opens downward (e.g., ).

• Increasing/Decreasing: Determined by the sign of the leading coefficient and the degree.

Combining and Composing Power Functions

• Combining: Adding, subtracting, or multiplying power functions yields new functions.

• Composition: means plugging one function into another.

• Example: If and , then .

Power Equations

A power equation is an equation where the variable appears as a base with an exponent, such as .

• Solving: Take the appropriate root of both sides, e.g., .

• Calculator Use: For non-integer exponents or roots, calculators can be used to approximate solutions.

• Kepler’s Third Law: In astronomy, relates the period and radius of planetary orbits: .

Root Functions

Root functions involve expressions like .

• Solving: Isolate the root and raise both sides to the appropriate power.

• Example: .

Transformations on Graphs

Transformations shift or change the shape of graphs.

• Vertical Shift: shifts up/down.

• Horizontal Shift: shifts right/left.

• Reflection: reflects over the x-axis.

• Stretch/Compression: stretches (|a| > 1) or compresses (0 < |a| < 1).

• Example: is shifted right 2 units and up 3 units.

Piecewise Functions

A piecewise function is defined by different expressions over different intervals of the domain.

• Notation:

• Evaluating: Determine which piece applies for a given x-value.

• Example: For above, , .