2.5 Shortcut: Vertex formula

In previous chapter, we learned all about factoring quadratic equations. There are a lot of applications of quadratic equations like calculating possible profits, computing for area. There are a lot of interesting uses of quadratic equations too, and there’s an interesting article about 101 uses of quadratic equation. Now in this chapter, we will get to learn more about Quadratic functions especially on its characteristics, and its general form.

We will learn in this chapter that Quadratic Functions have a parabolic graph. The parabola may open down, or may open up. The peak of the parabola is called the vertex. If the parabola is opening down, then the vertex is the highest point or the maximum. If the parabola opens down then the vertex is the lowest point or the minimum. The line that passes through that vertex is called the axis of symmetry. It also has an x intercept, where y=0, and the y intercept where x=0.A quadratic equation also have a domain and a range. The domain is simply the x values, and the range is the y values.

We will also learn more on the general form of the quadratic function, f(x)=ax2+bx+cf(x) = ax^2 + bx + c in Not every quadratic function that you would come across to would look the same, and there would be times that you would need to transform them into the general form to make solving them easier.

Apart from learning about the characteristics and the general form of quadratic function, we will also learn how to graph a quadratic function and at the same time learn how to identify the equation used in the graph given.

In previous chapters, we had discussed Quadratic Equations, starting with understanding their characteristics, knowing how their graph looks like, familiarizing ourselves with the general form and learning how the different factoring methods work.

Given that we have already established those concepts, we will now learn how to solve quadratic equations. We could always resort to the easiest way of solving them by using a free quadratic formula calculator, but it still pays to know how to solve them manually.

In this chapter we will learn the three ways to solve a quadratic equation. The first method is through factoring given that the quadratic that we have is factorable, otherwise, the method is not applicable.

We spent some time on discussing about factoring quadratics in previous chapters, we it's expected that you already understand this method. If the quadratic is in its general form, which is ax2+bx+c=0ax^2 + bx + c = 0, then we simply need to factor the quadratic, and solve for the variable given like in the case of x2+5x+6x^2 + 5x + 6, we know that the factors are (x+2) and (x+3) which would lead us to the value of x, which are 2, and 3. If it isn’t in its general form then we need to rearrange it to make things simpler.

The second method is completing squares, which we had discussed in previous chapter. This method requires us to convert the quadratic equation we have from general form (ax2+bx+c=0)(ax^2 + bx + c = 0) to vertex form y=a(xh)2+ky = a(x - h)^2 + k in order to proceed.

The last method we will learn about is using the quadratic formula, x=[b±(b24ac)]2ax = \frac{[-b\pm\sqrt{(b^2-4ac)}]}{2a}. The b24acb^2-4ac is referred to as the discriminant, which would tell us the nature of the roots. The roots can be rational, irrational or imaginary. This is simply a plug and play method where you use the general form, ax2+bx+c=0ax^2 + bx + c = 0 to get the values you need to substitute in the formula.

Shortcut: Vertex formula

Lessons

  • 2.
    Converting general form into vertex form by applying the vertex formula
    Convert each quadratic function from general form to vertex form by using the vertex formula.
    • b)
      y=3x260x50y = - 3{x^2} - 60x - 50
    • c)
      y=12x2+x52y = \frac{1}{2}{x^2} + x - \frac{5}{2}
    • d)
      y=5xx2y = 5x - {x^2}
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Shortcut: Vertex formula

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