sketch the graph of the following function?1.) y = -x² + 6x - 3 downward ?2.) y = 4x² + 12x + 9 up?3.) y = -2x² + 8x - 11 down?​

Here’s a step-by-step explanation for sketching the graphs of the given quadratic functions, including their direction (upward/downward), vertices, intercepts, and key features:   1. y = -x^2 + 6x - 3 (Opens Downward) Direction: Downward (coefficient of x^2 is negative).Vertex: x = -\frac{b}{2a} = -\frac{6}{2(-1)} = 3 Plug x = 3 into the equation: y = -(3)^2 + 6(3) - 3 = -9 + 18 - 3 = 6 Vertex: (3, 6) . Intercepts: - Y-intercept: x = 0 \Rightarrow y = -3 .- X-intercepts: Solve -x^2 + 6x - 3 = 0 : x = \frac{-6 \pm \sqrt{36 - 12}}{-2} = 3 \pm \sqrt{6} Approximate roots: x \approx 0.55 and x \approx 5.45 . Graph: - Opens downward with vertex at (3, 6) .- Crosses the x-axis at 3 \pm \sqrt{6} and the y-axis at (0, -3) .   2. y = 4x^2 + 12x + 9 (Opens Upward) Direction: Upward (coefficient of x^2 is positive).Vertex: x = -\frac{b}{2a} = -\frac{12}{2(4)} = -1.5 Plug x = -1.5 into the equation: y = 4(-1.5)^2 + 12(-1.5) + 9 = 9 - 18 + 9 = 0 Vertex: (-1.5, 0) . Intercepts: - Y-intercept: x = 0 \Rightarrow y = 9 .- X-intercepts: The vertex is on the x-axis ( y = 0 ), so the parabola touches the x-axis at x = -1.5 (a perfect square: y = (2x + 3)^2 ). Graph: - Opens upward with vertex at (-1.5, 0) .- Touches the x-axis at x = -1.5 and crosses the y-axis at (0, 9) .   3. y = -2x^2 + 8x - 11 (Opens Downward) Direction: Downward (coefficient of x^2 is negative).Vertex: x = -\frac{b}{2a} = -\frac{8}{2(-2)} = 2 Plug x = 2 into the equation: y = -2(2)^2 + 8(2) - 11 = -8 + 16 - 11 = -3 Vertex: (2, -3) . Intercepts: - Y-intercept: x = 0 \Rightarrow y = -11 .- X-intercepts: Solve -2x^2 + 8x - 11 = 0 :Discriminant: b^2 - 4ac = 64 - 88 = -24 .No real roots (does not cross the x-axis). Graph: - Opens downward with vertex at (2, -3) .- Does not cross the x-axis (negative discriminant).- Crosses the y-axis at (0, -11) .   Key Features of All Graphs 1. Axis of Symmetry: Vertical line through the vertex ( x = \text{vertex x-coordinate} ).2. Direction: Determined by the sign of x^2 -term.3. Vertex: Highest/lowest point of the parabola.4. Intercepts: Points where the graph crosses the axes. Let me know if you’d like a visual.

Answered by kyleshane1011 | 2025-07-30