Graph a quadratic function

Learning Objectives Identify the key features of a parabola (vertex, axis of symmetry, intercepts, concavity) from its equation. Graph a quadratic function given in standard form, f(x) = ax^2 + bx + c. Graph a quadratic function given in vertex form, f(x) = a(x-h)^2 + k. Determine the vertex of a parabola algebraically using the formula x = -b/(2a). Verify the vertex of a parabola by finding where the first derivative equals zero (f'(x) = 0). Determine the domain and range of a quadratic function from its graph and equation. Accurately sketch the graph of a quadratic function by plotting its vertex and intercepts. Ever wondered about the perfect arc of a basketball shot or the path of a satellite? 🏀 That's a parabola, the beautiful curve described by a quadratic...

TermDefinitionExample ParabolaThe distinctive U-shaped curve that represents a quadratic function on a graph.The graph of y = x^2 is a simple parabola that opens upwards with its lowest point at the origin (0,0). VertexThe highest or lowest point on a parabola. It is the turning point of the graph.For the parabola y = (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 y = 2x^2 - 4x + 1, the vertex is at (1, -1), so the axis of symmetry is the vertical line x = 1. Roots (or Zeros)The x-coordinates of the points where the parabola intersects the x-axis. These are the solutions to the equation f(x) = 0.The roots of f(x) = x^2 - 4 are x = 2 and x = -2, so the x-intercepts are...

Standard Form f(x) = ax^2 + bx + c The most common form of a quadratic function. The y-intercept is directly given by 'c'. The concavity is determined by the sign of 'a'. 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). Be mindful that the formula has 'x-h', so the sign of h is often opposite of what it appears. Vertex Formula (Algebraic) x_{vertex} = -\frac{b}{2a} Used with the standard form to find the x-coordinate of the vertex. To find the y-coordinate, substitute this x-value back into the original function: y_{vertex} = f(-\frac{b}{2a}). Vertex Formula (Calculus) f'(x) = 0 The vertex is a stationary point where the slope of the tangent line is zero. Fi...