Characteristics of quadratic functions

Learning Objectives Identify the vertex, axis of symmetry, intercepts, and direction of opening from any form of a quadratic function. Determine the domain, range, and intervals of increase and decrease for any quadratic function. Convert a quadratic function between standard, vertex, and factored forms to analyze its properties. Apply the discriminant to determine the number and nature of the roots of a quadratic equation. Determine the maximum or minimum value of a quadratic function and interpret it in applied contexts. Connect the sign of the leading coefficient to the function's end behavior and overall shape. Sketch an accurate graph of a quadratic function by identifying its key characteristics. How does a satellite dish focus signals to one single point? 🛰️ Th...

TermDefinitionExample VertexThe highest or lowest point on the parabola, representing the function's maximum or minimum value. It is the point where the function changes direction from increasing to decreasing, or vice versa.For the function f(x) = (x - 3)^2 + 5, the vertex is at the point (3, 5). Axis of SymmetryA vertical line that passes through the vertex and divides the parabola into two mirror-image halves.For f(x) = (x - 3)^2 + 5, the axis of symmetry is the vertical line x = 3. Zeros (Roots or x-intercepts)The point(s) where the parabola intersects the x-axis. These are the solutions to the equation f(x) = 0.For f(x) = x^2 - 4, the zeros are x = -2 and x = 2, corresponding to the points (-2, 0) and (2, 0). Domain and RangeThe domain is the set of all possible input values (x-...

Vertex Form f(x) = a(x - h)^2 + k This form directly reveals the vertex at (h, k). The value of 'a' determines the direction of opening (up if a > 0, down if a < 0) and the vertical stretch or compression. Standard Form & Vertex Formula f(x) = ax^2 + bx + c. The x-coordinate of the vertex is given by x = -b / (2a). This form directly reveals the y-intercept at (0, c). To find the vertex, calculate the x-coordinate using the formula, then substitute it back into the function to find the y-coordinate, f(-b / (2a)). The Discriminant Δ = b^2 - 4ac Used with the standard form f(x) = ax^2 + bx + c. If Δ > 0, there are two distinct real roots. If Δ = 0, there is exactly one real root (a repeated root). If Δ < 0, there are no real roots (two complex...