Determine the direction of the parabola.

Question

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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.
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.
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.
Find the axis of symmetry.
The x-coordinate of the axis of symmetry is given by the formula $x = -\frac{b}{2a}$ .
$x = -\frac{-3}{2(-2)} = -\frac{-3}{-4} = -\frac{3}{4}$
The axis of symmetry is the vertical line $x = -\frac{3}{4}$ . 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.
Find the vertex.
The x-coordinate of the vertex is $x = -\frac{3}{4}$ . Substitute this value back into the equation to find the y-coordinate of the vertex.
$y = -2\left(-\frac{3}{4}\right)^2 - 3\left(-\frac{3}{4}\right) + 1$ $y = -2\left(\frac{9}{16}\right) + \frac{9}{4} + 1$ $y = -\frac{18}{16} + \frac{9}{4} + 1$ $y = -\frac{9}{8} + \frac{18}{8} + \frac{8}{8}$ $y = \frac{-9 + 18 + 8}{8} = \frac{17}{8}$
The vertex is at $\left(-\frac{3}{4}, \frac{17}{8}\right)$ . Since $\frac{17}{8} = 2.125$ , 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 $y = -2x^2 - 3x + 1$ .

The correct option is (A).
$\boxed{\text{A}}$

Point	x	y
x-intercept	−1.78	0
Maximum	−0.75	2.13
y-intercept	0	1
x-intercept	0.28	0

Determine the direction of the parabola.

Answer

Step-by-step solution

Determine the direction of the parabola.

Find the y-intercept.

Find the axis of symmetry.

Find the vertex.

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