Determine the direction of the parabola.

To find the correct graph for the equation y = -2x^2 - 3x + 1, we need to analyze the properties of this quadratic function. Step 1: Determine the direction of the parabola. The equation is in the form y = ax^2 + bx + c. Here, a = -2, b = -3, and c = 1. Since a = -2 is negative, the parabola opens downwards. All given options show parabolas opening downwards, so this doesn't eliminate any options yet. Step 2: Find the y-intercept. The y-intercept occurs when x = 0. y = -2(0)^2 - 3(0) + 1 = 1 So, the graph must pass through the point (0, 1). All options appear to have a positive y-intercept. Step 3: Find the axis of symmetry. The x-coordinate of the axis of symmetry is given by the formula x = -(b)/(2a). x = -(-3)/(2(-2)) = -(-3)/(-4) = -(3)/(4) The axis of symmetry is the vertical line x = -(3)/(4). This means the axis of symmetry is to the left of the y-axis. • Option (A) shows the axis of symmetry to the left of the y-axis. • Option (B) shows the axis of symmetry to the right of the y-axis. This is incorrect. • Option (C) shows the axis of symmetry to the left of the y-axis. • Option (D) shows the axis of symmetry to the left of the y-axis. So, option (B) is eliminated. Step 4: Find the vertex. The x-coordinate of the vertex is x = -(3)/(4). Substitute this value back into the equation to find the y-coordinate of the vertex. y = -2(-(3)/(4))^2 - 3(-(3)/(4)) + 1 y = -2((9)/(16)) + (9)/(4) + 1 y = -(18)/(16) + (9)/(4) + 1 y = -(9)/(8) + (18)/(8) + (8)/(8) y = (-9 + 18 + 8)/(8) = (17)/(8) The vertex is at (-(3)/(4), (17)/(8)). Since (17)/(8) = 2.125, the vertex is above the x-axis. • Option (A) shows the vertex above the x-axis. This is consistent. • Option (C) shows the vertex below the x-axis. This is incorrect. • Option (D) shows the vertex on the x-axis. This is incorrect. Based on these steps, option (A) is the only graph that matches all the characteristics of the equation y = -2x^2 - 3x + 1. The correct option is (A). A That's 2 down. 3 left today — send the next one.