Graph the quadratic function below. $$ y=-2 x^{2}-4 x-4 $$

Question

UpStudy AI · Accepted Answer

It seems that there was an issue with analyzing the function $$ y = -2x^2 - 4x - 4 $$. However, I can still guide you through the process of graphing this quadratic function step by step.

### Step 1: Identify the Standard Form
The given quadratic function is in the standard form $$ y = ax^2 + bx + c $$, where:
- $$ a = -2 $$
- $$ b = -4 $$
- $$ c = -4 $$

### Step 2: Determine the Vertex
The vertex of a quadratic function can be found using the formula:
$$
x = -\frac{b}{2a}
$$
Substituting the values of $$ a $$ and $$ b $$:
$$
x = -\frac{-4}{2 \cdot -2} = \frac{4}{-4} = -1
$$
Now, substitute $$ x = -1 $$ back into the function to find the $$ y $$-coordinate of the vertex:
$$
y = -2(-1)^2 - 4(-1) - 4 = -2(1) + 4 - 4 = -2 + 4 - 4 = -2
$$
Thus, the vertex is at the point $$ (-1, -2) $$.

### Step 3: Determine the Axis of Symmetry
The axis of symmetry is the vertical line that passes through the vertex:
$$
x = -1
$$

### Step 4: Find the Y-Intercept
The y-intercept occurs when $$ x = 0 $$:
$$
y = -2(0)^2 - 4(0) - 4 = -4
$$
So, the y-intercept is at the point $$ (0, -4) $$.

### Step 5: Find the X-Intercepts
To find the x-intercepts, set $$ y = 0 $$:
$$
0 = -2x^2 - 4x - 4
$$
Dividing the entire equation by -2 gives:
$$
0 = x^2 + 2x + 2
$$
Now, we can use the quadratic formula:
$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$
Substituting $$ a = 1, b = 2, c = 2 $$:
$$
x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} = \frac{-2 \pm \sqrt{4 - 8}}{2} = \frac{-2 \pm \sqrt{-4}}{2} = \frac{-2 \pm 2i}{2} = -1 \pm i
$$
Since the discriminant is negative, there are no real x-intercepts.

### Step 6: Sketch the Graph
1. Plot the vertex at $$ (-1, -2) $$.
2. Draw the axis of symmetry at $$ x = -1 $$.
3. Plot the y-intercept at $$ (0, -4) $$.
4. Since there are no real x-intercepts, the parabola opens downwards (as $$ a < 0 $$) and does not cross the x-axis.

### Conclusion
The graph of the function $$ y = -2x^2 - 4x - 4 $$ is a downward-opening parabola with its vertex at $$ (-1, -2) $$ and a y-intercept at $$ (0, -4) $$. The parabola does not intersect the x-axis.

If you need a visual representation, you can use graphing software or tools to plot these points and sketch the parabola accordingly.
To graph the quadratic function $$ y = -2x^2 - 4x - 4 $$: 1. **Vertex**: At $$ (-1, -2) $$. 2. **Axis of Symmetry**: $$ x = -1 $$. 3. **Y-Intercept**: $$ (0, -4) $$. 4. **No X-Intercepts**. 5. **Parabola Opens Downwards**. Plot these points and sketch the downward-opening parabola accordingly.

Graph the quadratic function below. \( y=-2 x^{2}-4 x-4 \)

Solución de inteligencia artificial de Upstudy

Responder

Solución

Extra Insights

preguntas relacionadas

Latest Algebra Questions