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.
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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

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).
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