Characteristic Points Common Core II – Day 3

Warm Up # 2 a) Factor to solve the equation 3x2 – 16x + 5 = 0. b) Explain what the solutions tell you about the graph. c) Using what you know about the quadratic function, draw a rough sketch.

Hw Check

Quiz – You have 20 minutes

Essential Question (2.25.2014) How do I use algebraic reasoning to find the characteristic points (x-intercepts, vertex, axis of symmetry, y-intercept) of a quadratic function?

Use Doc Cam and Worksheet Summary: Have students fill out the graphic organizer on quadratics.

Finding Characteristics Points using Zeros Polynomial Factor the polynomial What are the Zeros Find the Average of the Zeros a) x2 + 8x + 15 = 0 b) x2 -13x + 42 = 0 c) x2 +2x – 24 = 0 2. Determine the x-intercepts for each polynomial above: (write each x-intercept as an ordered pair!) a)_____________________ b)________________________ c)_______________________

Look at the graph (using your graphing calculator) of each polynomial above. Determine the relationship between the “average of the zeros” and the Minimum (lowest point) of the graph.   Average the zeros to find the x value of your VERTEX!!

4. Given the equation x2 – 2x – 35 = 0, without looking at the graph, where would you expect the minimum to be located? 5. The minimum/maximum (the VERTEX) should always be written as an ORDERED PAIR. 6. But there are two numbers in an ordered pair. How could you find the y – value for the number 4? Substitute the value in for x, evaluate to find the y! 7. Find the ordered pair that matches the minimum from problem 4.

Let’s remember everything we know: Without using a calculator, it will make sketching a graph much easier! Try graphing the next problem without a calculator. x2 + 2x – 8 = 0 Let’s remember everything we know: Polynomial Factor: Zeros: Minimum: x2 + 2x – 8

8. Go back and find the Vertex of each polynomial from #1 8. Go back and find the Vertex of each polynomial from #1. Also state if it is a maximum or a minimum! a)___________________ b)___________________ c)___________________

What about the y-intercept? 7. 8. 9. 10. Using your graphing calculator graph (one-at-a-time) graph numbers 7-10 from your homework! Determine the y-intercept of each quadratic!

Y-intercept Where you cross the y-axis! It is when x is 0! In a quadratic it is the “c” value or the constant! If you forget this rule – you can always just substitute 0 in for x and solve to find the y- value of the y-intercept.

Guided Practice – Part A Find coordinates of x-intercepts, y-intercept, and maximum or minimum points on the graphs of these quadratic functions. f(x) = x (7 – x) f(x) = -x (7 – x) f(x) = (x-3) (x-8) f(x) = -2(x – 3)(x – 8)

Guided Practice – Part B Write rules for quadratic functions with graphs that meet these conditions: x-intercepts at (-2, 0) and (6, 0) with graph opening upward x-intercepts at (2, 0) and (6, 0) with minimum point at (4, -8) x-intercepts at (-2, 0) and (6, 0) with y-intercept at (0, -60) y=a(x+2)(x-6) y = 2(x – 2)(x – 6) Y = 5 (x+2) (x-6)

Extra Practice

Let’s Play some Musical Chairs! There are 5 stations around the classroom with a quadratic equation When I say go, you have 30 seconds to take your worksheet and pencil and move to one station Once everyone is at a station, the music will begin playing and you will need to spend the duration of the song trying to find the important key points When the music stops, you must stop writing immediately and go to the next station When we are finished with 5 songs, you will be given the opportunity to finish your graphs with one last song

Homework Extended Practice Worksheet Page 3, #9 (part d, f, g) #10 (a, c, e)

Finding Extrema Investigation 2 CP 2 – Page 332 CP

Homework # 7 and # 8

Homework