Question

Consider the function y=-\left(x+3\right)^2

Complete the following table of values.

x -5 -4 -3 -2 -1
y

Answer 1

[AI-generated answer — Bot GPT 3.5]

To complete the table of values, we substitute each given x-value into the function and evaluate for y.

For x = -5:
y = -(-5+3)^2
y = -(2)^2
y = -4

For x = -4:
y = -(-4+3)^2
y = -(1)^2
y = -1

For x = -3:
y = -(-3+3)^2
y = -(0)^2
y = 0

For x = -2:
y = -(-2+3)^2
y = -(1)^2
y = -1

For x = -1:
y = -(-1+3)^2
y = -(2)^2
y = -4

Thus, the completed table of values is:
x -5 -4 -3 -2 -1
y -4 -1 0 -1 -4

Answer 2

Plot the graph points

Answer 3

[AI-generated answer — Bot GPT 3.5]

To plot the graph points, we can use the x and y values from the completed table of values.

The points are: (-5, -4), (-4, -1), (-3, 0), (-2, -1), (-1, -4)

Plotting these points on a coordinate plane, we get the graph of the function:

Answer 4

What is the maximum y value?

Answer 5

[AI-generated answer — Bot GPT 3.5]

The function y = -(x+3)^2 is a downward opening parabola. Since the coefficient of the squared term is negative, the vertex of the parabola represents the maximum point.

To find the x-coordinate of the vertex, we set the derivative of the function to zero and solve for x:

dy/dx = -2(x+3) = 0
-2x - 6 = 0
-2x = 6
x = -3

Next, we substitute this x-value back into the original function to find the corresponding y-value:

y = -(x+3)^2
y = -(-3+3)^2
y = -(0)^2
y = 0

Therefore, the maximum y-value is 0.

Answer 6

What is the maximum x value?

Answer 7

[AI-generated answer — Bot GPT 3.5]

Since the function y = -(x+3)^2 is a downward opening parabola, the maximum value of x would occur when y = 0. In other words, we are looking for the x-coordinate of the vertex of the parabola.

From the previous response, we found that the vertex occurs at x = -3. Therefore, the maximum x-value is -3.

Answer 8

What are the coordinates of the vertex? State your answer in the form \left(a,b\right).

Answer 9

[AI-generated answer — Bot GPT 3.5]

The vertex of the parabola y = -(x+3)^2 is given by the coordinates (a, b), where a represents the x-coordinate and b represents the y-coordinate.

In this case, we have already determined that the x-coordinate of the vertex is -3. To find the corresponding y-coordinate, we substitute this x-value back into the original function:

y = -(x+3)^2
y = -(-3+3)^2
y = -(0)^2
y = 0

Therefore, the coordinates of the vertex are (-3, 0).