The function you've provided is $f(x) = 3x^2 - 2x + 5$. This is a quadratic function, which typically forms a parabola when graphed.

Here are a few key properties and steps you might want to explore:

1. Vertex: The vertex of the parabola can be found using the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients from the general form of a quadratic function $ax^2 + bx + c$.

   • For $f(x) = 3x^2 - 2x + 5$, $a = 3$ and $b = -2$.
   • The x-coordinate of the vertex is $x = -\frac{-2}{2(3)} = \frac{2}{6} = \frac{1}{3}$.
   • To find the y-coordinate, substitute $x = \frac{1}{3}$ back into the function: $f\left(\frac{1}{3}\right) = 3\left(\frac{1}{3}\right)^2 - 2\left(\frac{1}{3}\right) + 5 = 3\left(\frac{1}{9}\right) - \frac{2}{3} + 5 = \frac{1}{3} - \frac{2}{3} + 5 = \frac{-1}{3} + 5 = \frac{14}{3}$
   • So, the vertex is at $\left(\frac{1}{3}, \frac{14}{3}\right)$.

2. Axis of Symmetry: This is the vertical line that passes through the vertex, which is $x = \frac{1}{3}$.

3. Y-Intercept: The y-intercept is the point where the function crosses the y-axis, which occurs when $x = 0$.

   • $f(0) = 3(0)^2 - 2(0) + 5 = 5$.
   • So, the y-intercept is at $(0, 5)$.

4. Direction of Opening: Since the coefficient of $x^2$ (which is 3) is positive, the parabola opens upwards.

5. Discriminant: The discriminant of a quadratic equation $ax^2 + bx + c$ is given by $\Delta = b^2 - 4ac$. It helps determine the nature of the roots.

   • For this function, $\Delta = (-2)^2 - 4(3)(5) = 4 - 60 = -56$.
   • Since the discriminant is negative, the equation has two complex roots and no real roots.

Would you like further details on any of these steps, or do you have any other questions?

Here are some related questions you might find helpful:

1. How do you find the roots of a quadratic function?
2. What is the significance of the discriminant in quadratic equations?
3. How can you derive the equation of a parabola given its vertex and a point on the curve?
4. What are the steps to complete the square for the function $f(x) = 3x^2 - 2x + 5$?
5. How does the coefficient of $x^2$ affect the shape of the parabola?

Tip: Always check the discriminant when analyzing a quadratic function, as it provides valuable information about the nature of the roots.