$$ \begin{array}{l} ext { Consider the following function. Complete parts (a) through (e) } \ ext { below. } \ y=x^{2}+12 x+20 \ ext { a. Determine if the parabola whose equation is given opens } \ ext { upward or downward. Choose the correct answer below. } \ ext { The parabola opens downward. } \ ext { The parabola opens upward. } \ ext { bind the vertex. } \ ext { The vertex is } \square . \ ext { (Type an ordered pair.) }\end{array} $$.... (Ty.

Question

$$ \begin{array}{l}	ext { Consider the following function. Complete parts (a) through (e) } \ 	ext { below. } \ y=x^{2}+12 x+20 \ 	ext { a. Determine if the parabola whose equation is given opens } \ 	ext { upward or downward. Choose the correct answer below. } \ 	ext { The parabola opens downward. } \ 	ext { The parabola opens upward. } \ 	ext { bind the vertex. } \ 	ext { The vertex is } \square . \ 	ext { (Type an ordered pair.) }\end{array} $$.... (Ty.

UpStudy AI · Accepted Answer

To analyze the function $$ y = x^{2} + 12x + 20 $$, we will complete the following steps:

### Part (a): Determine the direction of the parabola
The direction in which a parabola opens is determined by the coefficient of the $$ x^{2} $$ term.

1. If the coefficient is positive, the parabola opens upward.
2. If the coefficient is negative, the parabola opens downward.

In our case, the coefficient of $$ x^{2} $$ is $$ 1 $$ (which is positive), so the parabola opens upward.

### Part (b): Find the vertex
The vertex of a parabola given in the standard form $$ y = ax^2 + bx + c $$ can be found using the formula for the x-coordinate of the vertex:

$$
x = -\frac{b}{2a}
$$

Where:
- $$ a = 1 $$
- $$ b = 12 $$

Let's calculate the x-coordinate of the vertex.

1. Substitute $$ a $$ and $$ b $$ into the formula:

$$
x = -\frac{12}{2 \cdot 1}
$$

Now, let's calculate this value.
Calculate the value by following steps:
- step0: Calculate:
  $$-\frac{12}{2\times 1}$$
- step1: Calculate:
  $$\frac{12}{2\times 1}$$
- step2: Rewrite the expression:
  $$\frac{12}{2}$$
- step3: Reduce the fraction:
  $$6$$
- step4: Reduce the fraction:
  $$-6$$
The x-coordinate of the vertex is $$ -6 $$.

Next, we need to find the y-coordinate of the vertex by substituting $$ x = -6 $$ back into the original function:

$$
y = (-6)^{2} + 12(-6) + 20
$$

Let's calculate this value.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(-6\right)^{2}+12\left(-6\right)+20$$
- step1: Multiply the numbers:
  $$\left(-6\right)^{2}-72+20$$
- step2: Evaluate the power:
  $$36-72+20$$
- step3: Calculate:
  $$-16$$
The y-coordinate of the vertex is $$ -16 $$.

### Conclusion
The vertex of the parabola is at the point $$ (-6, -16) $$.

### Summary of Answers:
- The parabola opens **upward**.
- The vertex is $$ (-6, -16) $$.
The parabola opens upward and the vertex is at (-6, -16).

…
(Ty.

Upstudy AI Solution

Answer

Solution

Part (a): Determine the direction of the parabola

Part (b): Find the vertex

Conclusion

Summary of Answers:

Bonus Knowledge

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… (Ty.