To find the minimum value of the quadratic function $f(x) = 2x^2 + 18x + 46$, we can use the vertex formula. The vertex of a quadratic function $ax^2 + bx + c$ is given by the formula:

$x = -\frac{b}{2a}$

Here, $a = 2$ and $b = 18$. Plugging in the values:

$x = -\frac{18}{2(2)} = -\frac{18}{4} = -4.5$

Now, substitute $x = -4.5$ back into the function to find the minimum value:

$f(-4.5) = 2(-4.5)^2 + 18(-4.5) + 46$

Let's calculate the exact value.The minimum value of the function $f(x) = 2x^2 + 18x + 46$ occurs at $x = -4.5$, and the minimum value is $5.5$.

If you have any further questions or would like more details, feel free to ask!

Here are 5 related questions to explore:

1. How would you determine the axis of symmetry for this quadratic function?
2. What are the x-intercepts of the function, if any?
3. How does the coefficient of $x^2$ affect the direction of the parabola?
4. Can you find the y-intercept of the function?
5. How would the graph of this function compare to the graph of $f(x) = 2x^2$?

Tip: The vertex of a parabola represents either the maximum or minimum point depending on the direction of the parabola. If the parabola opens upwards, the vertex is the minimum point, and if it opens downwards, it's the maximum point.