2.1Quadratic Functions

OBJECTIVESSWBAT: 1) write a quadratic function given two points 2) Graph a quadratic function by finding the vertex and intercepts 3) Convert a quadratic function from standard form into vertex form Lanugage Objective:SWBAT write in complete sentences what the parts of the vertex represent for a given word problem.

The parent function for quadratics is y = x2. This means that all other quadratics are some variation of this basic equation.

The vertex form of a quadratic is:

In this equation the vertex is at (h,k)

Ex: State the vertex of the following quadratic:

vertex is (5,2)

Given the following graph, write a quadratic equation in vertex form.Note: The vertex is at (1,2) and the point (2,0) is also on the graph.Plug in the extra point for x and y

Given the following graph, write a quadratic equation in vertex form.Note: The vertex is at (2,0) and the point (3,2) is also on the graph.Plug in the extra point for x and y

Many quadratics are not given to us in vertex form, but instead in standard form:

Ex: Write an equation for a quadratic that has x-intercepts at 4 and 8.x-ints written as factorsFOIL / DistributeCombine like terms

Now, if you are given this form you must be able to sketch graphs.

To sketch a parabola you NEED at least the vertex, x-intercepts, and y-intercept to graph it accurately.

To find the vertex:

To find the x-intercepts: FACTOR = 0

To find the y-intercept: plug in 0 for x and solve!

Ex: Sketch the graph after finding all necessary points:Vertex:So, vertex is at (-1,0)X-intercepts:X-intercepts are -1 and -1 (dont need to write the same one twice)y-intercepts:So, y-int at 1Now we can sketch the graph

So, vertex is at (-1,0)X-intercepts are -1So, y-int at 1Just plot the vertex and intercepts then connect the dots!

Ex: Sketch the graph after finding all necessary points:Vertex:So, vertex is at (-6,-16)X-intercepts:X-intercepts are -10 and -2y-intercepts:So, y-int at 20Now we can sketch the graph

So, vertex is at (-6,-16)X-intercepts are -10,-2So, y-int at 20Just plot the vertex and intercepts then connect the dots!

Ex: Just find the x-intercepts for this one for practice!Since this does not factor nicely, we have to use the quadratic formula To graph these, put them in the calc to get a decimal approx: xs = -2.5 and -7.4

The profit P (in dollars) for Ahmads record business is

where x is the number of songs sold on iTunes.

How many songs will he have to sell to yield a maximum profit?

** maximum is the same as the vertex**So Ahmad would have to sell 350,000 songs to make the maximum possible profit.

What is his maximum possible profit?

The last thing you need to be able to do is covert from standard to vertex form.

In order to do this we have to complete the square

Ex: Put the following into vertex form:

Move non x to other sideFactor out the front numberSet up to complete a perfect squareDistribute the 2 before adding To the other side.Simply left, factor the rightGet y by itself again to finish!

Move non x to other sideFactor out the front numberSimply left, factor the rightGet y by itself again to finish!