Quadratic Function

Brainstorm Stylin’ Both are quadratics (parabolas) Not one-to-one (not invertible) Parent function is x^2 Both are positive Both are continuous One goes through the origin Polynomial Both go through at least two quadrants Passes vertical line test and fails the horizontal line test

Advanced Algebra 2 – Unit 2 10/20/2011 AGENDA DO NOW: Quadratic or NO? Look to the right of the board Agenda: Portfolio Recap More Quadratic VOCAB Think Pair Share FOILING Quadratics We will: Analyze the value and consequence of “a” coefficients Determine the role does “b” play Determine the vertex – MAX/MIN Calculate SOLUTIONS, roots, intercepts & zeros

Quadratic Function (y = ax 2 + bx + c) a, b, and c are called the coefficients. The graph will form a parabola. Each graph will have either a maximum or minimum point. There is a line of symmetry which will divide the graph into two halves.

y = x 2 a = 1, b = 0, c = 0 Minimum point (0,0) Axis of symmetry x=0 y=x 2

What happen if we change the value of a and c ? y=3x 2 y=-3x 2 y=4x 2 +3 y=-4x 2 -2

Recap (y = ax 2 +bx+c) When a is positive, When a is negative, When c is positive When c is negative the graph concaves UPWARD. happy the graph concaves downward. sad. the graph moves up c units. the graph moves down c units.

Quadratic Function (y = ax 2 + bx + c) a, b, and c are called the coefficients. The graph will form a parabola. Each graph will have either a maximum or minimum point. There is a line of symmetry which will divide the graph into two halves.

Let’s investigate MAX and MIN y=x 2 -4 y=x 2 +2x-15 y=-x 2 +5y=-x 2 -1

What do you notice about max/min and line of symmetry? Think pair share (2min) y=x 2 -4 y=x 2 +2x-15 y=-x 2 +5y=-x 2 -1

VERTEX: –b/2a, f(-b/2a) y=x 2 -4 y=x 2 +2x-15 y=-x 2 +2x -15 y=-2x 2 -x +4 Work with a friend – be ready to present!

? Explore cfm?method=cResource.dspView&Reso urceID=154 cfm?method=cResource.dspView&Reso urceID=154 Describe the changes in your own words.

Solving Quadratic Functions (ax 2 + bx + c = 0) Since y = ax 2 + bx +c, by setting y=0 we set up a quadratic equation. To find the solutions means we need to find the x-intercept(s). X-intercepts are also called ROOTS To make your life more complicated, they are also called ZEROS

What are x intercepts also called?

Solving Quadratic Functions (ax 2 + bx + c = 0) We know what a parabola looks like, so how many solutions or roots or zeros or x-intercepts can there be?? Think Pair and share out (3 minutes)

Find the Solutions y=x 2 -4 y=x 2 +2x-15 y=-x 2 +5y=-x 2 -1

Find the solutions y=x 2 +2x+1 y=-x 2 +4x-1

Observations Sometimes there are two solutions. Sometimes there is only one solution. Sometimes there is no solution at all…well…there are imaginary solutions…you are going to love them

To solve quadratic equations (graphing method) X 2 - 2x = 0 We could put y = x 2 -x into a calculator or sketch it to find x intercepts. This one has two solutions, x=0 and x=2. y=x 2 -2x

Another Method to find ROOTS? By factoring…let’s get it started

Other Methods By factoring…let’s get it started By using the quadratic formula

The End