**Quadratics** are **Second-Order Polynomials**. which take on the form:

In this form:

- \(x\) represents the independent variable
- \(y\) represents the dependent variable

A quadratic is referred to as a second-order polynomial since the highest exponent of \(x\) in a non-zero term is 2.

When a quadratic equation is represented visually, it is referred to as a parabola.

A quadratic function can be expressed algebraically in a few different ways:

**Standard Form**: \(y = ax^2 + bx+ c\)**Vertex Form**: \(y = a(x - h)² + k\)**Factored Form**: \(y = a(x - r)(x - s)\)

- The sign of \(a\) (+ or -) will determine which direction the parabola will open
- If \(a\) is positive \( (a > 0) \), the parabola will open upwards; if \(a\) is negative \( (a < 0) \), the parabola will open downwards

- Isn't constant unlike a Linear Function
- Pattern can be described as positive, zero, negative.

- The magnitude of \(|a|\) will determine whether the parabola will be
**stretched**or**compressed** - If \(|a|\) > 1, the parabola will be stretched, or will appear more narrow
- If \(|a|\) < 1, the parabola will be compressed, or will appear more wide

- Represent the
**roots**,**zeroes**, or**solutions**of a quadratic equation - Points where the parabola crosses the x-axis
- Can be determined algebraically by setting \(y = 0\) in the quadratic equation

- The point where the parabola crosses the y-axis
- Can be determined algebraically by setting \(x = 0\) in the quadratic equation

This point acts as either the **minimum** or **maximum point** of a quadratic function.

The characteristics of the vertex depend on the sign of the \(a\) value in the quadratic equation:

- If \(a\) > 0 or the parabola opens upward, the vertex will act as a minimum point
- If \(a\) < 0 or the parabola opens downward, the vertex will act as a maximum point

- The vertical line the goes through the vertex
- Determines the x-coordinate of the vertex
- Can be determined algebraically by dividing the sum of the x-intercepts by 2

- The hoizontal line that goes through the vertex
- It determines the y-coordinate of the vertex
- Can be determined algebraically by using the Axis of Symmetry as the x-value in the quadratic equation

Identify if the quadratic function \(x^2 + 3\) has x-intercepts and/or y-intercepts.

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