Characteristics of quadratic functions

Learning Objectives Identify the vertex of a parabola from its equation in standard or vertex form. Determine the equation of the axis of symmetry for a quadratic function. Calculate the x-intercepts (roots/zeros) and the y-intercept of a quadratic function. Determine the direction of opening (up or down) and identify the maximum or minimum value. State the domain and range of a quadratic function. Sketch a graph of a parabola using its key characteristics. Ever wondered about the perfect arc of a basketball shot or the curve of a satellite dish? 🏀 That's a parabola, and today we're learning its secrets! This tutorial will guide you through the key features, or characteristics, of quadratic functions. Understanding these properties allows us to graph parabolas ac...

TermDefinitionExample ParabolaThe distinctive U-shaped curve created by graphing a quadratic function. It can open upwards or downwards.The graph of the function f(x) = x^2 is a parabola that opens upwards with its lowest point at the origin (0,0). VertexThe highest or lowest point on the parabola. It is the 'turning point' of the curve.For f(x) = (x - 3)^2 + 5, the vertex is at the point (3, 5). Axis of SymmetryThe vertical line that passes through the vertex and divides the parabola into two perfect mirror images.For a parabola with a vertex at (3, 5), the axis of symmetry is the vertical line with the equation x = 3. Roots (or Zeros, x-intercepts)The point(s) where the parabola crosses the horizontal x-axis. At these points, the value of the function is zero (f(x) = 0). A par...

Standard Form and its Clues f(x) = ax^2 + bx + c Use this form to quickly find the y-intercept and direction of opening. The y-intercept is always at (0, c). If 'a' is positive, the parabola opens up. If 'a' is negative, it opens down. Vertex Form and its Clues f(x) = a(x - h)^2 + k This form directly gives you the vertex at the point (h, k). The value of 'a' tells you the direction of opening, just like in standard form. Axis of Symmetry Formula x = -b / (2a) When a quadratic is in standard form (ax^2 + bx + c), this formula gives you the x-coordinate of the vertex, which is also the equation for the axis of symmetry. To find the y-coordinate of the vertex, plug this x-value back into the function.