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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

Graph representing an parabola (or quadratic equation).

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!!


Review these lessons:
Try these questions:

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

Graph outlining the different characteristics of a parabola, including intercepts, vertex, axis, and optimal point.

Identify if the Quadratic Equation \(y=x^2 + 3\) has \(x\)-intercepts and/or \(y\)-intercepts.

We can make this process easier for ourselves by first creating a table of values to identify if there any intercepts:

x Values -3 -2 -1 0 1 2 3
y Values 12 7 4 3 4 7 12

Based on this table, we can determine that there aren't any \(x\)-intercepts, as there aren't any cells where \(y = 0\). However, we can determine that there is a \(y\)-intercept of \(\boldsymbol{3}\), since \(y = 3\) when \(x = 0\).

To further confirm our results, we can graph this quadratic to get a better idea of how it looks visually. We will learn other techniques for finding \(x\)-intercepts soon:

Graph of a parabola representing the quadratic equation y=x^2+3.
Try these questions: