Question

Consider the function y equals 3 x squared plus 12 x minus 23.
​(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 graph of the function is at the point
  
enter your response here. ​(Type an ordered​ pair.)
Part 2
​(b) Choose the correct graph below. Each graph shown below has a window of ​[minus​10, ​10] by ​[minus​40, ​10].
A.



A coordinate system has a horizontal x-axis from negative 10 to 10 in increments of 1 and a vertical y-axis from negative 40 to 10 in increments of 5. An upward U-shaped curve has a minimum in the third quadrant and passes through the positive and negative x-axis.
B.



A coordinate system has a horizontal x-axis from negative 10 to 10 in increments of 1 and a vertical y-axis from negative 40 to 10 in increments of 5. A downward U-shaped curve has a maximum on the positive x-axis and exits the viewing window in the third and fourth quadrants.
C.



A coordinate system has a horizontal x-axis from negative 10 to 10 in increments of 1 and a vertical y-axis from negative 40 to 10 in increments of 5. An upward U-shaped curve has a minimum on the negative y-axis and passes through the positive and negative x-axis.
D.

Answer 1

To analyze the function \( y = 3x^2 + 12x - 23 \), we can find the vertex and graph it.

### Part 1: Finding the Vertex

The vertex of a quadratic function represented in the form \( y = ax^2 + bx + c \) can be found using the formula:

\[
x = -\frac{b}{2a}
\]

For the given function:
- \( a = 3 \)
- \( b = 12 \)

Plugging in the values:

\[
x = -\frac{12}{2(3)} = -\frac{12}{6} = -2
\]

Now, we can substitute \( x = -2 \) back into the function to find the corresponding \( y \)-coordinate:

\[
y = 3(-2)^2 + 12(-2) - 23
\]
\[
y = 3(4) - 24 - 23
\]
\[
y = 12 - 24 - 23 = 12 - 47 = -35
\]

Thus, the vertex of the graph of the function is at the point \((-2, -35)\).

### Part 2: Select the Correct Graph

Since the function has a positive coefficient for \( x^2 \) (\( a = 3 > 0 \)), it will be a **U-shaped** parabola, which means the vertex is a **minimum** point.

Given the coordinates of the vertex \((-2, -35)\), we check the description of the graph:

- **Graph A**: Upward U-shaped curve, minimum in the third quadrant. (This fits our vertex being at \((-2, -35)\).)
- **Graph B**: Downward U-shaped curve. (Does not fit.)
- **Graph C**: Upward U-shaped curve, minimum on the negative y-axis. (This does not specify if it matches the vertex location.)
- **Graph D**: (No description provided.)

Based on the details, **Graph A** is most likely the correct graph since it matches the characteristics of the quadratic function and the vertex location.

### Final Answers

**Part 1:**
The vertex of the graph of the function is at the point \( (-2, -35) \).

**Part 2:**
The correct graph is **A**.