To find the vertex and axis of symmetry in a quadratic function, we start with the basic form of a quadratic equation:

$f(x) = ax^2 + bx + c.$

The vertex is the highest or lowest point of the curve. It depends on whether the curve opens up or down. We can find the $x$-coordinate of the vertex using this formula:

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

After we get the $x$-coordinate, we plug it back into the original function to find the $y$-coordinate. This gives us the vertex as a point: $(x, f(x))$.

Example: Let’s take the function $f(x) = 2x^2 + 4x + 1$. Here, $a = 2$ and $b = 4$.

First, we calculate the $x$-coordinate of the vertex:

$x = -\frac{4}{2(2)} = -\frac{4}{4} = -1.$

Now, we substitute $x = -1$ back into the function:

$f(-1) = 2(-1)^2 + 4(-1) + 1 = 2(1) - 4 + 1 = -1.$

So, the vertex is $(-1, -1)$.

Next, the axis of symmetry is a vertical line that goes through the vertex. We can write the equation for the axis of symmetry like this:

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

For our example, the axis of symmetry is simply the line $x = -1$.

To wrap it up, here’s how to find the vertex and axis of symmetry:

1. Use $x = -\frac{b}{2a}$ to find the $x$-coordinate of the vertex.
2. Substitute this value into the quadratic equation to get the $y$-coordinate.
3. The axis of symmetry is the line $x = -\frac{b}{2a}$.

By understanding these ideas, you can really improve your knowledge of quadratic functions!