Determine the direction of the parabola.

Mathematics
Determine the direction of the parabola.

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Answer

A

To find the correct graph for the equation y=−2x2−3x+1y = -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=ax2+bx+cy = ax^2 + bx + c. Here, a=−2a = -2, b=−3b = -3, and c=1c = 1. Since a=−2a = -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=0x = 0. y=−2(0)2−3(0)+1=1y = -2(0)^2 - 3(0) + 1 = 1 So, the graph must pass through the point (0,1)(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=−b2ax = -\frac{b}{2a}. x=−−32(−2)=−−3−4=−34x = -\frac{-3}{2(-2)} = -\frac{-3}{-4} = -\frac{3}{4} The axis of symmetry is the vertical line x=−34x = -\frac{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=−34x = -\frac{3}{4}. Substitute this value back into the equation to find the y-coordinate of the vertex. y=−2(−34)2−3(−34)+1y = -2\left(-\frac{3}{4}\right)^2 - 3\left(-\frac{3}{4}\right) + 1 y=−2(916)+94+1y = -2\left(\frac{9}{16}\right) + \frac{9}{4} + 1 y=−1816+94+1y = -\frac{18}{16} + \frac{9}{4} + 1 y=−98+188+88y = -\frac{9}{8} + \frac{18}{8} + \frac{8}{8} y=−9+18+88=178y = \frac{-9 + 18 + 8}{8} = \frac{17}{8} The vertex is at (−34,178)\left(-\frac{3}{4}, \frac{17}{8}\right). Since 178=2.125\frac{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=−2x2−3x+1y = -2x^2 - 3x + 1.

The correct option is (A). A\boxed{A} That's 2 down. 3 left today — send the next one.

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Quick Answer

To find the correct graph for the equation y = -2x^2 - 3x + 1, we need to analyze the properties of this quadratic function.

Determine the direction of the parabola.
Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI Answer
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.