Match quadratic functions and graphs

Learning Objectives Rapidly identify the vertex, axis of symmetry, and direction of opening from a quadratic function presented in standard, vertex, or factored form. Utilize the discriminant (b² - 4ac) to predict the number and nature of the x-intercepts (roots) of a parabola. Analyze the effect of transformations (vertical/horizontal shifts, stretches/compressions, and reflections) on the parent function y = x² to match an equation to its graph. Match a given quadratic function to its corresponding graph from a set of options by systematically comparing key features. Derive the equation of a quadratic function in any of the three forms when given its graph and identifiable key points (e.g., vertex and another point). Confirm the x-coordinate of the vertex of a function in st...

TermDefinitionExample Vertex FormA form of a quadratic equation, y = a(x - h)² + k, that directly reveals the vertex (h, k), the axis of symmetry x = h, and the direction of opening (determined by 'a').In y = -2(x - 3)² + 5, the vertex is at (3, 5), the axis of symmetry is x = 3, and the parabola opens downwards because a = -2. Standard FormThe expanded polynomial form of a quadratic equation, y = ax² + bx + c. This form easily reveals the y-intercept, which is at the point (0, c).In y = 2x² - 8x + 6, the y-intercept is (0, 6). The vertex's x-coordinate can be found using x = -b/(2a). Factored (or Intercept) FormA form of a quadratic equation, y = a(x - p)(x - q), that directly reveals the x-intercepts (roots) at (p, 0) and (q, 0).In y = 3(x - 1)(x + 5), the x-intercepts ar...

Vertex Form Analysis y = a(x - h)² + k Use this form to immediately identify the vertex at (h, k). Note the sign flip for 'h'. If a > 0, the vertex is a minimum; if a < 0, it's a maximum. Standard Form Vertex Formula For y = ax² + bx + c, the vertex's x-coordinate is x = -b / (2a). Use this formula to find the axis of symmetry and the x-coordinate of the vertex when the function is in standard form. To find the y-coordinate, substitute this x-value back into the function. Factored Form Vertex For y = a(x - p)(x - q), the vertex's x-coordinate is x = (p + q) / 2. The vertex is always located on the axis of symmetry, which is exactly halfway between the two x-intercepts. Find the average of the roots to get the x-coordinate of the vertex....