When you're learning about quadratic equations in Year 8 Maths, it's important to know the difference between two main forms: the standard form and the vertex form. Quadratic equations are expressions that look like this:

$y = ax^2 + bx + c$

Here, $a$, $b$, and $c$ are numbers (constants). Let's break down the differences between these two forms and how to change one into the other!

Standard Form

The standard form of a quadratic equation is:

$y = ax^2 + bx + c$

Key Points:

1. Coefficients:

   • The number $a$ is called the leading coefficient. It shows which way the parabola (a U-shaped graph) opens. If $a$ is greater than 0, it opens up. If $a$ is less than 0, it opens down.
   • The number $b$ helps find where the vertex (the highest or lowest point) is and the line of symmetry.
   • The number $c$ tells us where the curve crosses the y-axis.

2. Shape of the Parabola:

   • The standard form doesn’t directly show the vertex, so you might need to do some extra work, like completing the square or using the quadratic formula, to find the vertex and line of symmetry.

3. Finding Roots:

   • You can use the standard form to quickly find the roots (the x-values where the graph touches the x-axis) using the quadratic formula:

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

Vertex Form

The vertex form of a quadratic equation looks like this:

$y = a(x - h)^2 + k$

Here, $(h, k)$ is the vertex of the parabola.

Key Points:

1. Vertex Location:

   • In this form, the vertex is easy to find! For example, if you have $y = 2(x - 3)^2 + 1$, the vertex is at the point $(3, 1)$.

2. Graphing:

   • It's easier to graph with the vertex form because it shows the vertex right away. You won't need extra calculations like you do with the standard form.

3. Transformation Insights:

   • The vertex form helps you understand how the basic quadratic function changes. The $(x - h)$ part shows a sideways shift, and the $k$ value shows a vertical shift.

Converting Between Forms

To convert from standard form to vertex form, you need to complete the square. Let’s see how this works with an example:

Example: Change $y = x^2 + 6x + 5$ to vertex form.

1. Identify coefficients:

   • Here, $a = 1$, $b = 6$, and $c = 5$.

2. Complete the square:

   • Start with the part that includes $x$: $x^2 + 6x$.
   • Take half of 6 (which is 3) and square it (3² = 9).
   • Add and subtract this number:

   $y = (x^2 + 6x + 9) - 9 + 5$

   • Now simplify:

   $y = (x + 3)^2 - 4$

3. Write in vertex form:

   • Now we can write it as

$y = 1(x + 3)^2 - 4$

• The vertex is at the point $(-3, -4)$.

Conclusion

In conclusion, both standard and vertex forms have their own strengths when working with quadratic equations. The standard form is good for quickly finding roots using the quadratic formula, while the vertex form is great for graphing and easily showing where the vertex is. With a bit of practice, switching between these forms will help you understand quadratics better. So, when you face a quadratic equation next time, remember these differences and you'll be ready to tackle it!