Concept 3: Graphing Quadratic Functions & Concept 4: Characteristics of Quadratic Functions

Section A: Identify the vertex of each graph; identify whether it is a minimum or a maximum. Graph 1: A downward-facing parabola with vertex at (-1, 2). Handwritten note: "Vertex: (-1, 2) minimum" Graph 2: An upward-facing parabola with vertex at (2, 4). Handwritten note: "Vertex: (2, 4) maximum" Key Features of a Parabola Direction of Opening: When a > 0, the parabola opens up. When a < 0, the parabola opens down. Width: When |a| < 1, the parabola is wider than y = x^2 (vertical shrink). When |a| = 1, the parabola is the same width as y = x^2. When |a| > 1, the parabola is narrower than y = x^2 (vertical stretch). Vertex: The highest or lowest point of the parabola, located on the axis of symmetry. To find the vertex, plug in x = -b / 2a and solve for y. This yields a point (x, y). Axis of Symmetry: A vertical line passing through the vertex. x-intercepts: Where the parabola crosses the x-axis. Plug in y = 0 and solve for x. y-intercept: Where the parabola crosses the y-axis. Plug in x = 0 and solve for y. Without graphing the quadratic functions, complete the requested information: 3. f(x) = 3x^2 - 7x + 1 Direction of opening: up Vertex: min Wider or narrower than y = x^2: narrower 4. g(x) = -(5/4)x^2 + x - 3 Direction of opening: down Vertex: max Wider or narrower than y = x^2: narrower 5. y = (2/3)x^2 - 11 Direction of opening: up Vertex: min Wider or narrower than y = x^2: wider 6. y = -0.6x^2 + 4.3x - 9.1 Direction of opening: down Vertex: max Wider or narrower than y = x^2: wider Section 1: How to find the Vertex Step 1: List a = ___, b = ___, c = ___ Step 2: Find x = -b / 2a Step 3: Plug this x-value into the function (table) Step 4: This point is the vertex of the parabola Section 2: Graphing Step 1: Put the vertex you found in the center of your x-y chart. Step 2: Choose 2 x-values less than and 2 x-values more than your vertex. Step 3: Plug in these x values to get 4 more points. Step 4: Graph all 5 points.   Examples: 7. h(x) = 2x² + 4x + 1 Vertex: x = -b / 2a = -4 / (2 * 2) = -1 Vertex: (-1, -1) Max or min: min Direction of opening: up Domain: all real numbers R Range: y ≥ -1 Increasing: x > -1 Decreasing: x < -1 Comparison to y = x²: vertical stretch by a factor of 2, translation left 1 and down 1 8. k(x) = 2 - x - 1/2x² Written in standard form: k(x) = -1/2x² - x + 2 Vertex: x = -b / 2a = -(-1) / (2 * -1/2) = -1 Vertex: (-1, 2.5) Max or min: max Direction of opening: down Domain: R Range: y ≤ 2.5 Increasing: x < -1 Decreasing: x > -1 Comparison to y = x²: reflection in the x-axis, vertical shrink by a factor of 1/2, translation left 1 and up 2.5 9. State whether the function y = -3x² + 12x - 6 has a minimum value or a maximum value. Then find the minimum or maximum value.   Maximum value x = -b / 2a = -12 / (2 * -3) = 2 y = -3(2)² + 12(2) - 6 = 6 The maximum value is 6. Equation: f(x) = a(x - h)² + k Description: The graph of y = a(x - h)² + k is the parabola y = ax² translated Right h units and up/down k units. (opposite direction) The vertex is (h, k). The axis of symmetry is x = h. The graph opens up if a > 0 and down if a < 0. Examples: 10. y = (x + 1)² - 2 Vertex: (-1, -2) Max or min: min Direction of opening: up Axis of symmetry: x = -1 Domain: R Range: y ≥ -1 Increasing: x > -1 Decreasing: x < -1 Comparison to the graph of y = x²: translation left 1 and down 2. 11. y = -1/3(x - 1)² + 5 Vertex: (1, 5) Max or min: max Direction of opening: down Axis of symmetry: x = 1 Domain: R Range: y ≤ 5 Increasing: x < 1 Decreasing: x > 1 Comparison to the graph of y = x²: reflection in the x-axis, vertical shrink by a factor of 1/3, translation right 1 and up 5. 12. Graph y = -2(x - 1)(x - 5) Vertex: (3, 8) X-intercepts: (1, 0) and (5, 0) Max or min: max Direction of opening: down Axis of symmetry: x = 3 Domain: IR Range: y ≤ 8 Increasing: x < 3 Decreasing: x > 3 Compare to the graph of y = x²: reflection in the x-axis, vertical stretch by a factor of 2, translation right 3 and up 8. 13. y = -2(x - 1)(x + 3) Vertex: (-1, 8) X-intercepts: (1, 0) and (-3, 0) Max or min: max Direction of opening: down Axis of symmetry: x = -1 Domain: IR Range: y ≤ 8 Increasing: x < -1 Decreasing: x > -1 Compare to the graph of y = x²: reflection in the x-axis, vertical stretch by a factor of 2, translation left 1 and up 8.