To find the vertex and axis of symmetry of the quadratic function $f(x) = 2x^2 + 12x + 17$, we can use the following formulas:

1. Axis of Symmetry: The axis of symmetry of a quadratic function in the form $f(x) = ax^2 + bx + c$ is given by the formula: $x = \frac{-b}{2a}$
2. Vertex: The x-coordinate of the vertex is the value of the axis of symmetry, and we can substitute this value into the original function to find the y-coordinate.

Step-by-step process:

1. Identify coefficients:

   • $a = 2$
   • $b = 12$
   • $c = 17$

2. Find the axis of symmetry: Using the formula $x = \frac{-b}{2a}$: $x = \frac{-12}{2(2)} = \frac{-12}{4} = -3$ So, the axis of symmetry is $x = -3$.

3. Find the vertex: The x-coordinate of the vertex is $-3$, so now we substitute $x = -3$ into the original equation to find the y-coordinate of the vertex: $f(-3) = 2(-3)^2 + 12(-3) + 17 = 2(9) - 36 + 17 = 18 - 36 + 17 = -1$ So, the vertex is $(-3, -1)$.

Conclusion:

• Vertex: $(-3, -1)$
• Axis of symmetry: $x = -3$

Thus, the correct answer is A:
Vertex: (-3, -1), Axis of symmetry: $x = -3$.

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Would you like any further details or explanations? Here are 5 related questions to explore:

1. How can we convert a quadratic function into vertex form?
2. How do the values of $a$, $b$, and $c$ affect the graph of the quadratic function?
3. What is the significance of the vertex in the context of the quadratic function?
4. How can the discriminant be used to determine the nature of the roots of a quadratic equation?
5. What happens to the axis of symmetry and vertex when the coefficient $a$ changes sign?

Tip: The vertex always lies on the axis of symmetry, and the axis of symmetry divides the parabola into two mirror-image halves.