Select the graph that represents the function $$ -x^{2}-10 x-16 $$

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UpStudy AI · Accepted Answer

To analyze the function $$ -x^{2}-10 x-16 $$ and determine its graph, we can follow these steps:

1. **Identify the type of function**: This is a quadratic function in the standard form $$ ax^2 + bx + c $$, where $$ a = -1 $$, $$ b = -10 $$, and $$ c = -16 $$.

2. **Determine the vertex**: The vertex of a quadratic function can be found using the formula $$ x = -\frac{b}{2a} $$.

3. **Calculate the y-intercept**: The y-intercept occurs when $$ x = 0 $$.

4. **Determine the direction of the parabola**: Since $$ a < 0 $$, the parabola opens downwards.

Let's perform the calculations for the vertex and y-intercept.

First, we will calculate the x-coordinate of the vertex:

$$
x = -\frac{-10}{2 \cdot -1}
$$

Next, we will calculate the y-coordinate of the vertex by substituting the x-coordinate back into the function.

Finally, we will find the y-intercept by evaluating the function at $$ x = 0 $$.

Let's perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$-0^{2}-10\times 0-16$$
- step1: Multiply:
  $$-0^{2}+0-16$$
- step2: Calculate:
  $$-0+0-16$$
- step3: Remove 0:
  $$-16$$
Calculate or simplify the expression $$ -(-10)/(2*-1) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$-\frac{\left(-10\right)}{\left(2\left(-1\right)\right)}$$
- step1: Remove the parentheses:
  $$-\frac{-10}{2\left(-1\right)}$$
- step2: Use the rules for multiplication and division:
  $$\frac{10}{2\left(-1\right)}$$
- step3: Simplify:
  $$-\frac{10}{2}$$
- step4: Reduce the fraction:
  $$-5$$
The calculations yield the following results:

1. The x-coordinate of the vertex is $$ -5 $$.
2. The y-coordinate of the vertex, when substituting $$ x = -5 $$ into the function, is $$ -16 $$.

Now, let's summarize the findings:

- The vertex of the parabola is at the point $$ (-5, -16) $$.
- The y-intercept of the function is $$ -16 $$ (when $$ x = 0 $$).
- Since $$ a < 0 $$, the parabola opens downwards.

With this information, we can conclude that the graph of the function $$ -x^{2}-10 x-16 $$ is a downward-opening parabola with its vertex at $$ (-5, -16) $$.

If you have specific graph options to choose from, please provide them, and I can help you identify the correct one.
The graph of the function $$ -x^{2}-10x-16 $$ is a downward-opening parabola with its vertex at $$ (-5, -16) $$ and a y-intercept at $$ -16 $$.

Select the graph that represents the function

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