Question

Consider the function y equals 3 x squared plus 18 x minus 8.
​(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 on the negative y-axis 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 curve rises from left to right at a decreasing rate, passing through the negative x-axis. A second curve is to the right of the first curve and falls at an increasing rate through the positive x-axis.
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. A downward U-shaped curve has a maximum on the positive x-axis and exits the viewing window in the third and fourth quadrants.
D.

Answer 1

To find the vertex of the quadratic function \( y = 3x^2 + 18x - 8 \), we can use the formula for the vertex of a quadratic equation given in standard form \( ax^2 + bx + c \). The x-coordinate of the vertex can be calculated using the formula:

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

For our function, \( a = 3 \) and \( b = 18 \).

1. **Calculate the x-coordinate of the vertex**:

\[
x = -\frac{18}{2 \cdot 3} = -\frac{18}{6} = -3
\]

2. **Find the y-coordinate by substituting \( x = -3 \) back into the function**:

\[
y = 3(-3)^2 + 18(-3) - 8
\]
\[
y = 3(9) - 54 - 8
\]
\[
y = 27 - 54 - 8 = -35
\]

Thus, the coordinates of the vertex are \((-3, -35)\).

### Part (a)

The vertex of the graph of the function is at the point

\[
\boxed{(-3, -35)}
\]

### Part (b)

Next, we need to identify the correct graph among the provided options. The vertex \((-3, -35)\) is below the x-axis, and since the leading coefficient (3) is positive, the parabola opens upwards.

Given the descriptions of the graphs:
- **Graph A**: Describes an upward U-shaped curve with its minimum on the negative y-axis.
- **Graph B**: Describes two different curves, likely not a single quadratic.
- **Graph C**: Describes a downward U-shaped curve.
- **Graph D**: No description provided.

Based on these observations, **Graph A** is the only one that correctly represents an upward-opening parabola and the vertex being situated at \((-3, -35)\).

### Final Answer to Part (b)

The correct graph is **Graph A**.