Quadratic Functions
Lessons in this chapter
About Quadratic Functions
Quadratic Functions introduces the parabola, the distinctive U-shaped graph of any quadratic function, through three different equation forms.
Key Features of a Quadratic Graph has kids identify the vertex, axis of symmetry, and intercepts directly from a graph, before Intercept (or Factored) Form and Vertex Form cover two equation forms that make specific features: the x-intercepts, or the vertex, readable directly from the equation itself.
Transformations covers sliding, flipping, and stretching a parabola the same way Functions: Transformations covered any function's graph, and Modelling closes the chapter by writing a quadratic function to represent a real situation, like the path of a thrown object.
Each of the five lessons here opens as an Algebra 1 worksheet in the page: Key Features of a Quadratic Graph, Intercept (or Factored) Form, Vertex Form, Transformations and Modelling. Every question is scored as you answer it, so you find out on the spot whether a vertex has been read off vertex form with the sign of h the right way round. The practice is free to try, with no login needed for your first 3.
Frequently asked questions
Key Features of a Quadratic Graph names the parts of a parabola: the vertex, the axis of symmetry, the intercepts and whether it opens up or down. Intercept (or Factored) Form and Vertex Form then show how each written form puts a different feature in plain sight, the x-intercepts in one and the vertex in the other, and Transformations shifts, reflects and stretches the basic parabola. Modelling closes the chapter with real situations, such as the path of a thrown ball, that a quadratic describes. The chapter sits in the high school Functions conceptual category (F).
A parabola is the U-shaped graph of a quadratic function, and its vertex is the single highest or lowest point on that graph: the minimum value if the parabola opens upward, or the maximum if it opens downward.
Intercept form writes a quadratic as y = a(x - p)(x - q), where p and q are the x-intercepts (zeros) of the function, readable directly from the equation without any extra calculation.
Vertex form writes a quadratic as y = a(x - h)² + k, where (h, k) is the vertex, readable directly from the equation.
The axis of symmetry is the vertical line that passes through the vertex. From vertex form, it’s simply x = h; from standard form, it can be calculated as x = -b/2a.
The same shift, reflection, and stretch rules from Functions: Transformations apply directly, vertex form actually shows these transformations built into the equation itself, since h and k represent horizontal and vertical shifts from the parent function y = x².