Quadratic Functions
Lessons in this chapter
About Quadratic Functions
Quadratic Functions rebuilds the parabola from every angle: its graph, its equation forms, and the transformations that move it around the plane.
Key Features of a Quadratic Graph identifies the vertex, axis of symmetry, and intercepts directly from a parabola’s graph, before Intercept (or Factored) Form and Vertex Form cover the two equation forms that make those same features easy to read algebraically.
Transformations shows how changing a quadratic’s equation shifts, stretches, or reflects its graph, while Focus of a Parabola introduces the geometric definition of a parabola in terms of a focus point and directrix.
Modelling closes the chapter by using quadratic functions to represent and solve real situations, like projectile motion or area problems.
Every lesson in this chapter opens as an Algebra 2 worksheet you work in the page, from Key Features of a Quadratic Graph through Vertex Form to Modelling. The quadratic practice problems are scored as you answer them, so a misread vertex or a missed intercept shows up at that question rather than at the end. All six lessons are free and nothing has to be set up first.
Frequently asked questions
The chapter starts with Key Features of a Quadratic Graph, reading the vertex, axis of symmetry, and intercepts straight off a parabola. Intercept (or Factored) Form and Vertex Form then give the two equations that make those same features easy to read algebraically, Transformations shows how changing the equation shifts, stretches, or reflects the graph, and Focus of a Parabola adds the geometric definition behind the curve. Modelling closes with real situations such as projectile motion. This sits in the Algebra (A) and Functions (F) conceptual categories of the high school Common Core, with the focus and directrix work in Geometry (G).
The vertex, the axis of symmetry (the vertical line through the vertex), the direction it opens, and its x- and y-intercepts. These describe a parabola’s shape and position completely.
Intercept (or factored) form, y = a(x - p)(x - q), directly shows the x-intercepts p and q: set y to 0 and each factor gives one intercept immediately.
Vertex form, y = a(x - h)² + k, directly shows the vertex (h, k), no extra calculation needed to locate it.
The value a stretches or compresses the parabola vertically (and flips it if negative), while h and k in vertex form shift it horizontally and vertically.
A parabola's focus is a fixed point such that every point on the parabola is equidistant from the focus and a fixed line (the directrix). This geometric definition is what makes a parabola a parabola, beyond just its equation.