Select the graph that represents the function: $$ x^{2}+6 x-27 $$

Question

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To select the graph that represents the function $$ x^{2} + 6x - 27 $$, we first need to analyze the function.

1. **Identify the type of function**: This is a quadratic function in the standard form $$ ax^2 + bx + c $$, where $$ a = 1 $$, $$ b = 6 $$, and $$ c = -27 $$.

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

3. **Calculate the vertex**: Let's calculate the x-coordinate of the vertex.

4. **Determine the y-coordinate of the vertex**: Substitute the x-coordinate back into the function to find the y-coordinate.

5. **Identify the direction of the parabola**: Since $$ a > 0 $$, the parabola opens upwards.

Let's perform the calculations for the vertex.

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

$$
x = -\frac{b}{2a} = -\frac{6}{2 \cdot 1}
$$

Now, let's calculate it.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{-6}{\left(2\times 1\right)}$$
- step1: Remove the parentheses:
  $$\frac{-6}{2\times 1}$$
- step2: Calculate:
  $$-\frac{6}{2\times 1}$$
- step3: Rewrite the expression:
  $$-\frac{6}{2}$$
- step4: Reduce the fraction:
  $$-3$$
The x-coordinate of the vertex is $$ x = -3 $$.

Next, we will find the y-coordinate of the vertex by substituting $$ x = -3 $$ back into the function $$ x^{2} + 6x - 27 $$:

$$
y = (-3)^{2} + 6(-3) - 27
$$

Now, let's calculate this expression.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(-3\right)^{2}+6\left(-3\right)-27$$
- step1: Multiply the numbers:
  $$\left(-3\right)^{2}-18-27$$
- step2: Evaluate the power:
  $$9-18-27$$
- step3: Subtract the numbers:
  $$-36$$
The y-coordinate of the vertex is $$ y = -36 $$.

Thus, the vertex of the function $$ x^{2} + 6x - 27 $$ is at the point $$(-3, -36)$$.

### Summary of the function characteristics:
- **Vertex**: $$(-3, -36)$$
- **Direction**: The parabola opens upwards.

Now, you can look for a graph that has a vertex at $$(-3, -36)$$ and opens upwards.
The graph of the function $$ x^{2} + 6x - 27 $$ is a parabola opening upwards with its vertex at $$(-3, -36)$$.

Select the graph that represents the function:

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Answer

Solution

Summary of the function characteristics:

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