The Quadratic Formula is a really helpful tool in Year 11 Maths that helps us solve quadratic equations.

A quadratic equation looks like this: $ax^2 + bx + c = 0$. Here, $a$, $b$, and $c$ are numbers that don’t change, and $x$ is what we’re trying to find. The Quadratic Formula is written like this:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$$

We mainly use this formula to find the roots, also known as x-intercepts, of the quadratic equation. But it can also help us figure out the highest or lowest points of the quadratic function, especially when we graph it as a parabola.

Understanding Maxima and Minima

When we graph a quadratic equation, we get a shape called a parabola. Depending on the value of $a$:

• If $a > 0$, the parabola opens upward, and there is a lowest point, called the minimum, at the vertex.
• If $a < 0$, the parabola opens downward, and there is a highest point, called the maximum, at the vertex.

The vertex is an important point because it shows us the highest or lowest value of the quadratic function. To find this vertex, we use a special formula for its x-coordinate. The x-coordinate of the vertex, $x_v$, can be found with this formula:

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

Finding the Vertex and Identifying Extrema

Let’s go through an example. Look at this quadratic function:

$$f(x) = 2x^2 - 8x + 6.$$

Here, $a = 2$, $b = -8$, and $c = 6$. First, we find the x-coordinate of the vertex:

$$x_v = -\frac{-8}{2 \times 2} = \frac{8}{4} = 2.$$

Now, let’s find the y-coordinate of the vertex by plugging $x_v$ back into the function:

$$f(2) = 2(2)^2 - 8(2) + 6 = 8 - 16 + 6 = -2.$$

So, the vertex of the parabola is at the point $(2, -2)$. Since $a = 2 > 0$, we see that this is a minimum point. This means that the lowest value of the quadratic function is $-2$, and it happens when $x = 2$.

Conclusion

To sum it up, the Quadratic Formula helps us not only find the roots of a quadratic equation but also locate the vertex, which shows us the highest or lowest point of the quadratic function. Knowing where the vertex is can help us understand how the function looks when we graph it.

So, the next time you see a quadratic equation in your Year 11 Maths class, remember that the Quadratic Formula can give you great insights into how the function behaves. Keep up the great work in your learning!