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 \( b = 18 \).
- Calculate the x-coordinate of the vertex:
\[ x = -\frac{18}{2 \cdot 3} = -\frac{18}{6} = -3 \]
- 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.