Identifying the vertex of a quadratic function is an important skill that helps you draw quadratic equations well. A quadratic function usually looks like this:

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

In this expression, $a$, $b$, and $c$ are numbers. The number $a$ tells us which way the parabola opens. If $a$ is greater than 0, it opens up. If $a$ is less than 0, it opens down.

Step 1: Finding the Vertex

To find the vertex of the quadratic function, we can use this formula for the $x$-coordinate:

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

This formula helps us find the middle point by using some basic math. After you get the $x$-coordinate, you can plug that value back into the function to find the $y$-coordinate.

Example:

Let's look at this function:

$$f(x) = 2x^2 - 8x + 5$$

• Step 1: First, we need to identify $a$ and $b$.

  • Here, $a = 2$ and $b = -8$.

• Step 2: Now, let's calculate the $x$-coordinate of the vertex.

$$x = -\frac{-8}{2 \cdot 2} = \frac{8}{4} = 2$$

• Step 3: Next, substitute $x = 2$ back into the function to find $y$.

$$f(2) = 2(2)^2 - 8(2) + 5 = 8 - 16 + 5 = -3$$

So, the vertex is the point $(2, -3)$.

Step 2: Graphing the Vertex

Now that we have the vertex, we can mark this point on a graph. You can also figure out the axis of symmetry, which is a vertical line that passes through the vertex. This line can be defined by the same equation:

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

For our example, the axis of symmetry is the line $x = 2$.

Conclusion

With the vertex and the axis of symmetry, you can draw the parabola. You can also see if it opens upward or downward. Knowing how to find the vertex is really important for drawing any quadratic function correctly in your math class!