High School: Functions
Interpreting Functions F-IF.8a
a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
Students should be able to find the x-intercepts of a quadratic function using both factoring and completing the square. All that should be given to students is the standard form of a quadratic equation in the form y = ax2 + bx + c.
When a = 1, factoring is fairly easy. The equation can be factored into the form y = (x + p)(x + q), where p + q = b and pq = c. For example, if given the equation y = x2 – 9x + 18, we'd need p and q values such that p + q = -9 and pq = 18. A quick check will tell us that p = -3 and q= -6 are the values that make sense. So our factored equation is y = (x – 3)(x – 6).
Since y = 0 for x-intercepts, we can set the factored form of our equation to equal zero. The entire equation will be zero when either (or both) of the factors are zero. We can find the roots by solving each factor for x. The factors x – 3 = 0 and x – 6 = 0 mean that our x-intercepts are 3 and 6.
Completing the square is another way to factor. Starting with the standard form of the equation y = ax2 + bx + c, we can use the following steps to create a perfect square trinomial and solve for the x-intercepts that way.
1. Set y = 0.
2. Divide through by a (the coefficient in front of x2 must be 1).
3. Subtract the constant factor from both sides.
4. Rewrite the function in the form x2 + 2hx + h2.
5. Rewrite the function in the form (x + h)2.
6. Take the square root of both sides.
It turns out that completing the square is a straightforward way to derive the quadratic equation. Pretty sweet, right?
We can also describe extreme values of the function, or the vertex of the parabola. If we return to the original form of the equation, y = ax2 + bx + c, the coordinates of the vertex can be written as
If we look at the other form of the equation, y = (x + h)2, we can say that the centerline (or line of symmetry) lies at x = -h.
So much information can be extracted from the different forms of writing a quadratic equation. If students are lost and confused within the many terms of a quadratic equation, the standard form is their landmark. From there, they can get wherever they need to go and find whatever they need to find.