Unit 36: Parabolas and Factoring

Quadratic Equations and Parabolas

In the next several units, we will be looking at quadratic equations. A quadratic equation is just an equation that has an (x^2) and the highest exponent is 2. When we graph a quadratic equation, the shape of the graph is a parabola. A parabola is basically just a right-side-up or upside-down “U”- shape. The highest or lowest point on the parabola is the vertex. A vertical line of symmetry runs through the vertex.

The standard form of a parabola is: (y=ax^2+bx+c)

The a value determines how wide the parabola is and which direction the parabola opens. If a is positive, the parabola opens upward. If a is negative, the parabola opens downward. The c represents the y-intercept.

Here are two examples of parabolas:

(y=2x^2-4x-6)

  • The a value is positive, so the graph opens upward.
  • The c value is –6. This tells us the y-intercept.

(y=-x^2-x+6)

  • The a value is negative, so the graph opens downward.
  • The c value is 6. This tells us the y-intercept.
Factoring

We can factor the two equations above to find the x-intercepts (aka the zeros or solutions). To factor the equation on the left, we would first divide each term by 2 to get: (y=x^2-2x-3). When we factor, we want to figure out what two binomial terms we could multiply together to get our equation. We’ll look a little further into factoring strategies in the following topic.

Factored form will always look like:

$$y=(xpm h)(xpm k)$$

The key thing to remember with factoring is that when you add (h+k), it must equal the coefficient to x (–2 in the equation above). When you multiply (htimes k), it must equal the constant (–3 in the equation above). Therefore, h and k must equal (–3) and (1). So our equation in factored form would be:

$$y=(x-3)(x+1)$$

If we drew a graph of (y=(x-3)(x+1)), it would be exactly the same as the graph of (y=2x^2-4x-6) pictured above. The difference in the two equations is that we can easily figure out the x-intercepts if the equation is in factored form. The x-intercepts are the x coordinates when the y value is 0. Looking at the factored equation above, we can see that y will be 0 if x = 3 or x = -1. That means that the graph crosses over the x-axis at those two points.

Practice:

Practice factoring the following equations:

(y=x^2+8x+15)

(y=x^2+x-12)

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