Consider the function y equals 12 x minus 2 x squared.

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.

GPT-4o mini · Answer

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 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).