Question
Consider the function y equals 12 x minus 2 x squared.
(a) Give the coordinates of the vertex of the graph of the function.
(b) Graph the function on a window that includes the vertex.
Question content area bottom
Part 1
(a) The vertex of the function is
enter your response here. (Type an ordered pair.)
Part 2
(b) Choose the correct graph below.
A.
A coordinate system has a horizontal x-axis from negative 20 to 30 in increments of 2 and a vertical y-axis from negative 230 to 210 in increments of 20. An upward U-shaped curve passes through the points (2, negative 16), (3, negative 18), and (4, negative 16). All coordinates are approximate.
B.
A coordinate system has a horizontal x-axis from negative 20 to 30 in increments of 2 and a vertical y-axis from negative 230 to 210 in increments of 20. A downward U-shaped curve passes through the points (negative 1, negative 12), (0, 0), and (1, negative 12). All coordinates are approximate.
C.
A coordinate system has a horizontal x-axis from negative 20 to 30 in increments of 2 and a vertical y-axis from negative 230 to 210 in increments of 20. A downward U-shaped curve passes through the points (2, 16), (3, 18), and (4, 16). All coordinates are approximate.
D.
(a) Give the coordinates of the vertex of the graph of the function.
(b) Graph the function on a window that includes the vertex.
Question content area bottom
Part 1
(a) The vertex of the function is
enter your response here. (Type an ordered pair.)
Part 2
(b) Choose the correct graph below.
A.
A coordinate system has a horizontal x-axis from negative 20 to 30 in increments of 2 and a vertical y-axis from negative 230 to 210 in increments of 20. An upward U-shaped curve passes through the points (2, negative 16), (3, negative 18), and (4, negative 16). All coordinates are approximate.
B.
A coordinate system has a horizontal x-axis from negative 20 to 30 in increments of 2 and a vertical y-axis from negative 230 to 210 in increments of 20. A downward U-shaped curve passes through the points (negative 1, negative 12), (0, 0), and (1, negative 12). All coordinates are approximate.
C.
A coordinate system has a horizontal x-axis from negative 20 to 30 in increments of 2 and a vertical y-axis from negative 230 to 210 in increments of 20. A downward U-shaped curve passes through the points (2, 16), (3, 18), and (4, 16). All coordinates are approximate.
D.
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To analyze the function \( y = 12x - 2x^2 \), let's start with part (a) to find the coordinates of the vertex.
### Part 1: Finding the Vertex
The function can be written in the standard form of a quadratic, \( y = ax^2 + bx + c \).
Here, \( a = -2 \), \( b = 12 \), and \( c = 0 \).
The x-coordinate of the vertex (for a quadratic function in the form \( y = ax^2 + bx + c \)) can be found using the formula:
\[
x = -\frac{b}{2a}
\]
Plugging in our values:
\[
x = -\frac{12}{2 \cdot (-2)} = -\frac{12}{-4} = 3
\]
Next, we find the y-coordinate of the vertex by substituting \( x = 3 \) back into the original function:
\[
y = 12(3) - 2(3^2) = 36 - 18 = 18
\]
Thus, the vertex of the function is at the coordinates \( (3, 18) \).
**Answer for Part 1: (3, 18)**
---
### Part 2: Graphing the Function
Since the function \( y = 12x - 2x^2 \) is a downward-opening parabola (as indicated by the negative coefficient of \( x^2 \)), we know the vertex represents the highest point on the graph. The vertex we calculated (3, 18) falls within the indicated range of the graph, specifically between \( x = 0 \) and \( x = 30 \) on the x-axis and \( y = -230 \) to \( 210 \) on the y-axis.
From the options provided:
- **Option A** describes an upward U-shaped curve – which is incorrect.
- **Option B** describes a downward U-shape that does not match our vertex location.
- **Option C** describes a downward U-shaped curve that does not match our vertex location.
- **Option D** does not provide enough information directly in the prompt.
Given your question, the correct graph representing \( y = 12x - 2x^2 \), with a vertex at \( (3, 18) \), should depict a downward-opening parabola that peaks at this vertex.
Since none of the options explicitly contain the vertex (3, 18) and refer to a downward-opening shape, it seems that you would choose the graph closest to this point or indicate that the representation does not exist among provided choices.
If **Option B** is the closest in shape (a parabola opening downward), you might choose this option, but further assessment may be needed from the graphical options.
### Final Answers:
- Vertex: **(3, 18)**
- Graph: Choose the closest downward-opening graph if none directly represent (3, 18).
### Part 1: Finding the Vertex
The function can be written in the standard form of a quadratic, \( y = ax^2 + bx + c \).
Here, \( a = -2 \), \( b = 12 \), and \( c = 0 \).
The x-coordinate of the vertex (for a quadratic function in the form \( y = ax^2 + bx + c \)) can be found using the formula:
\[
x = -\frac{b}{2a}
\]
Plugging in our values:
\[
x = -\frac{12}{2 \cdot (-2)} = -\frac{12}{-4} = 3
\]
Next, we find the y-coordinate of the vertex by substituting \( x = 3 \) back into the original function:
\[
y = 12(3) - 2(3^2) = 36 - 18 = 18
\]
Thus, the vertex of the function is at the coordinates \( (3, 18) \).
**Answer for Part 1: (3, 18)**
---
### Part 2: Graphing the Function
Since the function \( y = 12x - 2x^2 \) is a downward-opening parabola (as indicated by the negative coefficient of \( x^2 \)), we know the vertex represents the highest point on the graph. The vertex we calculated (3, 18) falls within the indicated range of the graph, specifically between \( x = 0 \) and \( x = 30 \) on the x-axis and \( y = -230 \) to \( 210 \) on the y-axis.
From the options provided:
- **Option A** describes an upward U-shaped curve – which is incorrect.
- **Option B** describes a downward U-shape that does not match our vertex location.
- **Option C** describes a downward U-shaped curve that does not match our vertex location.
- **Option D** does not provide enough information directly in the prompt.
Given your question, the correct graph representing \( y = 12x - 2x^2 \), with a vertex at \( (3, 18) \), should depict a downward-opening parabola that peaks at this vertex.
Since none of the options explicitly contain the vertex (3, 18) and refer to a downward-opening shape, it seems that you would choose the graph closest to this point or indicate that the representation does not exist among provided choices.
If **Option B** is the closest in shape (a parabola opening downward), you might choose this option, but further assessment may be needed from the graphical options.
### Final Answers:
- Vertex: **(3, 18)**
- Graph: Choose the closest downward-opening graph if none directly represent (3, 18).
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.