To find the vertex and axis of symmetry for the function $f(x) = 5 - 6x + 3x^2$, we can rewrite it in the form $ax^2 + bx + c$. Here, $a = 3$, $b = -6$, and $c = 5$.

1. Find the Vertex:

The formula for the x-coordinate of the vertex is given by:

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

Substituting the values of $a$ and $b$:

$x = -\frac{-6}{2 \cdot 3} = \frac{6}{6} = 1$

Now, substitute $x = 1$ into the function to find the y-coordinate of the vertex:

$f(1) = 5 - 6(1) + 3(1)^2 = 5 - 6 + 3 = 2$

So, the vertex is at $(1, 2)$.

2. Find the Axis of Symmetry:

The axis of symmetry is a vertical line that passes through the vertex. Its equation is:

$x = 1$

Summary:

• Vertex: $(1, 2)$
• Axis of Symmetry: $x = 1$

Would you like more details or have any questions?

Here are some related questions you might find helpful:

1. How do you complete the square to find the vertex form of a quadratic function?
2. What is the effect of changing the coefficient $a$ on the graph of a quadratic function?
3. How do you determine if a quadratic function opens upwards or downwards?
4. How can you find the y-intercept of a quadratic function?
5. What are the applications of finding the vertex in real-life scenarios?

Tip: The vertex represents the maximum or minimum point of a parabola, which can be particularly useful in optimization problems.