Characteristics of quadratic functions

Learning Objectives Identify the vertex, axis of symmetry, and direction of opening of a parabola from its equation. Determine the domain and range of any quadratic function. Calculate the x-intercepts (roots/zeros) and y-intercept of a quadratic function. Analyze the effect of the parameters 'a', 'h', and 'k' in the vertex form f(x) = a(x - h)^2 + k. Convert a quadratic function from standard form to vertex form by completing the square. Sketch the graph of a quadratic function using its key characteristics. Ever wonder about the perfect arc of a basketball shot or the shape of a satellite dish? 🏀 That's the power of the parabola in action! This tutorial will dissect the anatomy of quadratic functions and their U-shaped graphs, called pa...

TermDefinitionExample ParabolaThe symmetrical, U-shaped curve that represents a quadratic function on a graph. 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 a parabola. It is the point where the parabola changes direction.For f(x) = (x - 3)^2 + 5, the vertex is at the point (3, 5). Since the parabola opens up, this is a minimum point. Axis of SymmetryThe vertical line that passes through the vertex and divides the parabola into two mirror-image halves.For the parabola with vertex (3, 5), the axis of symmetry is the vertical line x = 3. Zeros (or Roots/x-intercepts)The point(s) where the parabola intersects the x-axis. These are the solutions to the equat...

Standard Form f(x) = ax^2 + bx + c This form is useful for quickly identifying the y-intercept, which is the point (0, c). The sign of 'a' determines if the parabola opens upwards (a > 0) or downwards (a < 0). Vertex Form f(x) = a(x - h)^2 + k This form directly reveals the vertex of the parabola, which is at the point (h, k). The axis of symmetry is the line x = h. The sign of 'a' still determines the direction of opening. Axis of Symmetry from Standard Form x = -b / (2a) When a quadratic is in standard form, this formula allows you to calculate the x-coordinate of the vertex, which also gives you the equation for the axis of symmetry.