Consider The Function Y = − 2 ( X + 3 ) 2 + 7 Y = -2(x+3)^2 + 7 Y = − 2 ( X + 3 ) 2 + 7 .Complete The Table:${ \begin{tabular}{|l|l|} \hline X X X & Y Y Y \ \hline & \ \hline & \ \hline & \ \hline & \ \hline & \ \hline \end{tabular} }$Determine The Following

6. Consider the function $y = -2(x+3)^2 + 7$

Completing the Table and Determining Key Characteristics

Introduction

In this section, we will be working with a quadratic function in the form of $y = -2(x+3)^2 + 7$. Our goal is to complete a table with specific values of $x$ and their corresponding $y$ values. Additionally, we will determine key characteristics of the function, including its vertex, axis of symmetry, and the direction it opens.

Completing the Table

To complete the table, we will substitute specific values of $x$ into the function and calculate the corresponding $y$ values.

$x$	$y$
-6
-3
0
3
6

Let's start by substituting $x = -6$ into the function:

$y = -2(-6+3)^2 + 7$ $y = -2(-3)^2 + 7$ $y = -2(9) + 7$ $y = -18 + 7$ $y = -11$

So, the first row of the table is:

$x$	$y$
-6	-11

Next, let's substitute $x = -3$ into the function:

$y = -2(-3+3)^2 + 7$ $y = -2(0)^2 + 7$ $y = -2(0) + 7$ $y = 7$

So, the second row of the table is:

$x$	$y$
-6	-11
-3	7

Now, let's substitute $x = 0$ into the function:

$y = -2(0+3)^2 + 7$ $y = -2(3)^2 + 7$ $y = -2(9) + 7$ $y = -18 + 7$ $y = -11$

So, the third row of the table is:

$x$	$y$
-6	-11
-3	7
0	-11

Next, let's substitute $x = 3$ into the function:

$y = -2(3+3)^2 + 7$ $y = -2(6)^2 + 7$ $y = -2(36) + 7$ $y = -72 + 7$ $y = -65$

So, the fourth row of the table is:

$x$	$y$
-6	-11
-3	7
0	-11
3	-65

Finally, let's substitute $x = 6$ into the function:

$y = -2(6+3)^2 + 7$ $y = -2(9)^2 + 7$ $y = -2(81) + 7$ $y = -162 + 7$ $y = -155$

So, the fifth row of the table is:

$x$	$y$
-6	-11
-3	7
0	-11
3	-65
6	-155

Determining Key Characteristics

Now that we have completed the table, let's determine some key characteristics of the function.

Vertex

The vertex of a quadratic function in the form of $y = ax^2 + bx + c$ is given by the formula $x = -\frac{b}{2a}$. In our function, $a = -2$ and $b = 0$, so the vertex occurs at $x = -\frac{0}{2(-2)} = 0$.

To find the $y$-coordinate of the vertex, we substitute $x = 0$ into the function:

$y = -2(0+3)^2 + 7$ $y = -2(3)^2 + 7$ $y = -2(9) + 7$ $y = -18 + 7$ $y = -11$

So, the vertex of the function is at the point $(0, -11)$.

Axis of Symmetry

The axis of symmetry of a quadratic function in the form of $y = ax^2 + bx + c$ is given by the equation $x = -\frac{b}{2a}$. In our function, $a = -2$ and $b = 0$, so the axis of symmetry is given by the equation $x = -\frac{0}{2(-2)} = 0$.

Direction

The direction of a quadratic function in the form of $y = ax^2 + bx + c$ is determined by the sign of the coefficient $a$. In our function, $a = -2$, which is negative. Therefore, the function opens downward.

Conclusion

In this section, we completed a table with specific values of $x$ and their corresponding $y$ values for the function $y = -2(x+3)^2 + 7$. We also determined key characteristics of the function, including its vertex, axis of symmetry, and direction. The vertex of the function is at the point $(0, -11)$, the axis of symmetry is given by the equation $x = 0$, and the function opens downward.
6. Consider the function $y = -2(x+3)^2 + 7$

Q&A: Completing the Table and Determining Key Characteristics

Introduction

In the previous section, we completed a table with specific values of $x$ and their corresponding $y$ values for the function $y = -2(x+3)^2 + 7$. We also determined key characteristics of the function, including its vertex, axis of symmetry, and direction. In this section, we will answer some frequently asked questions about the function.

Q&A

Q: What is the vertex of the function?

A: The vertex of the function is at the point $(0, -11)$.

Q: What is the axis of symmetry of the function?

A: The axis of symmetry of the function is given by the equation $x = 0$.

Q: In which direction does the function open?

A: The function opens downward.

Q: How do I find the $y$-coordinate of the vertex?

A: To find the $y$-coordinate of the vertex, substitute $x = 0$ into the function.

Q: What is the value of the function at $x = -6$?

A: The value of the function at $x = -6$ is $-11$.

Q: What is the value of the function at $x = 3$?

A: The value of the function at $x = 3$ is $-65$.

Q: How do I determine the key characteristics of the function?

A: To determine the key characteristics of the function, use the following steps:

1. Find the vertex of the function by substituting $x = -\frac{b}{2a}$ into the function.
2. Find the $y$-coordinate of the vertex by substituting $x = -\frac{b}{2a}$ into the function.
3. Determine the axis of symmetry by using the equation $x = -\frac{b}{2a}$.
4. Determine the direction of the function by checking the sign of the coefficient $a$.

Conclusion

In this section, we answered some frequently asked questions about the function $y = -2(x+3)^2 + 7$. We also provided step-by-step instructions on how to determine the key characteristics of the function. By following these steps, you can easily determine the vertex, axis of symmetry, and direction of the function.

Additional Resources

• Quadratic Functions
• Vertex Form of a Quadratic Function
• Axis of Symmetry

Practice Problems

• Find the vertex of the function $y = 3(x-2)^2 + 1$.
• Find the $y$-coordinate of the vertex of the function $y = 2(x+1)^2 - 3$.
• Determine the axis of symmetry of the function $y = -4(x-1)^2 + 2$.
• Determine the direction of the function $y = 5(x+2)^2 - 1$.

Solutions

• The vertex of the function $y = 3(x-2)^2 + 1$ is at the point $(2, 1)$.
• The $y$-coordinate of the vertex of the function $y = 2(x+1)^2 - 3$ is $-3$.
• The axis of symmetry of the function $y = -4(x-1)^2 + 2$ is given by the equation $x = 1$.
• The function $y = 5(x+2)^2 - 1$ opens upward.