Intro to Quadratics

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

\(y = x^2\)

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: \(y = a(x - r)(x - s)\)

* We will cover these forms in later lessons!!

Characteristics of a Parabola

Direction of Opening

• 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

Slope

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

Stretch/Compression

• 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

x-intercepts

• 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

y-intercept

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

Vertex

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

Axis of Symmetry

• 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

Optimal Point

• 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

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