Lycée Jean-Piaget - Neuchâtel 2016-2017

Mathematics 2MPES Chapter 3

QuadraticFunctions2

We know now how to graph quadratic functions in standard form (or vertex form). OK, so far, so good! But you may have noticed a problem already, which is that most quadratic functions that we've dealt with in the past did not look like:

𝑦 = 𝑎 (𝑥 − 𝑘)! + ℎ .

They looked more like … well, you know:

𝑦 = 𝑎𝑥! + 𝑏𝑥 + 𝑐

or something like that.

Question: How do we graph that? Answer: We put it into the forms we now know how to graph. Question: OK, but how do we do that? Answer: Completing the square! First of all, you must know your Remarkable Identities. We’ll consider the perfect square for 𝑥 plus a constant or 𝑥 minus a constant. Let’s use 𝑎 for the constant. We have:

The perfect squares 𝒙+ 𝒂 𝟐 = 𝒙𝟐 + 𝟐𝒂𝒙+ 𝒂𝟐 𝒙− 𝒂 𝟐 = 𝒙𝟐 − 𝟐𝒂𝒙+ 𝒂𝟐

Now, allow me to demonstrate how to complete the square and pay a careful attention to the ways we will process.

Step Example The function itself 𝑦 = 𝑥! − 6𝑥 − 8 As usual, we start by putting the number (−8 in this case) on the other side.

𝑦 = 𝑥! − 6𝑥 − 8𝑦 + 8 = 𝑥! − 6𝑥

+8

If we look at the right part, we can see the beginning of a perfect square. In order to complete that square, we need to add 9.

𝑥! − 6𝑥𝑥! − 2𝑎𝑥

gives 2𝑎 = 6 so 𝑎 = 3

if 𝑎 = 3 then 𝑎! = 9 so 9 is needed to complete the square

So we add the number you need to complete the square (9 here), on both side of the equal sign.

𝑦 + 8 = 𝑥! − 6𝑥𝑦 + 8+ 9 = 𝑥! − 6𝑥 + 9𝑦 + 17 = 𝑥! − 6𝑥 + 9

+9𝐶𝐿𝑇

On the right side, we now have a perfect square and can rewrite it as such. Then, we isolate 𝑦 to come back to the function form (-17 here).

𝑦 + 17 = 𝑥 − 3 !

𝑦 = 𝑥 − 3 ! − 17−17

standard form

Here we are! Now you can graph the quadratic function

Vertex: 𝑉 = (3;−17), and it opens up

Your turn, now!

Lycée Jean-Piaget - Neuchâtel 2016-2017

Mathematics 2MPES Chapter 3

Exercise3.20𝑥! − 6𝑥 + 8

a. Complete the square using the process above to make it 𝑦 = 𝑎 (𝑥 − 𝑘)! + ℎ b. Find the vertex and the direction of opening, and draw a quick sketch

Exercise3.21𝑥! + 6𝑥 − 3

a. Complete the square using the process above to make it 𝑦 = 𝑎 (𝑥 − 𝑘)! + ℎ b. Find the vertex and the direction of opening, and draw a quick sketch

Exercise3.22𝑥! + 6𝑥 + 1

a. Complete the square using the process above to make it 𝑦 = 𝑎 (𝑥 − 𝑘)! + ℎ b. Find the vertex and the direction of opening, and draw a quick sketch

Exercise3.23𝑦 = 𝑥! + 2𝑥 + 5

a. Complete the square using the process above to make it 𝑦 = 𝑎 (𝑥 − 𝑘)! + ℎ b. Find the vertex and the direction of opening, and draw a quick sketch

Exercise3.24𝑦 = 𝑥! − 10𝑥 + 15

a. Complete the square using the process above to make it 𝑦 = 𝑎 (𝑥 − 𝑘)! + ℎ b. Find the vertex and the direction of opening, and draw a quick sketch