 Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane. WARM UP FEB. 10 TH XY

GRAPHING PROPERTIES OF QUADRATIC EQUATIONS

 This is the math term for the u-shape of a quadratic function.  Any quadratic function (one with an x 2 term), will have this same basic shape. PARABOLA

y = Ax 2 + Bx + C STANDARD FORM OF QUADRATIC FUNCTIONS

 Direction of opening- which way the open side of the parabola is facing.  y-intercept- where the graph crosses the y-axis. WHAT INFORMATION CAN I FIND FROM THE GRAPH OF A PARABOLA?

 Vertex- the point where your graph changes directions. (h, k) it is the same as your min or max value.  Axis of Symmetry- vertical line through the vertex that cuts the graph in half. WHAT INFORMATION CAN I FIND FROM THE GRAPH OF A PARABOLA?

 Maximum or minimum value- highest or lowest point of the graph  The max/min is the y-value of the vertex WHAT INFORMATION CAN I FIND FROM THE GRAPH OF A PARABOLA?

 Vertex: (1, 2)  The axis of symmetry is the vertical line that cuts the parabola in half. The equation of the AOS is the x-value of the vertex. WE CAN ALSO USE THE VERTEX TO FIND THE AXIS OF SYMMETRY AOS: Min. value:

 Direction of opening  Y-intercept  Vertex  AOS  Max/Min LET’S TAKE A LOOK AT A FEW PARABOLAS

DIRECTION OF OPENING, Y-INTERCEPT, VERTEX, AOS, MAX/MIN

 Find the y-intercept, direction of opening, vertex, AOS, and max/min value of the following function. Write these on your sticky note then put it on the board. STICKY NOTE PROBLEM

 What would the graph do if we expanded the view further left and right?  We use the parabola’s direction of the opening to see the end behavior.  Do the ends go up to infinity?  Or down to negative infinity? END BEHAVIOR

 The x-values that show where the parabola crosses the x-axis  We will find these values by graphing, factoring, or using the quadratic formula. ZEROS, ROOTS, X-INTERCEPTS

 Graphing: visually identify the intersections.  Factoring: factor the equation then set each set of parentheses equal to 0 and solve for x.  Quadratic Formula: plug A, B and C into the formula and simplify. ZEROS, ROOTS, X-INTERCEPTS

 If the parabola is completely above or below the x-axis, we say there are no real solutions  If the parabola sits on the x-axis, we say there is 1 real solution  If the parabola is on both sides of the x- axis (crosses twice), we say there are 2 real solutions NUMBER OF SOLUTIONS

IDENTIFY THE ZEROS IN THE GRAPHS

 Domain: the set of x values that exist on the function  If the graph were to squish to the x-axis, what values would be covered by the graph?  Range: the set of y-values that exist on the function  If the graph were to squish to the y-axis, what values would be covered by the graph? DOMAIN & RANGE

WHAT ARE THE DOMAIN & RANGE?

 Describe where the graphs are increasing and where they are decreasing INCREASING VS. DECREASING

 We can show the region of the graph that is increasing by an interval.  Intervals describe the range of x-values that meet the given requirement. INTERVALS

 We use interval notation to abbreviate the description.  List the starting and ending points of your interval, separated by a comma.  -  to -1 will look like: - , -1  Then we decide if there should be parentheses () or brackets []  Parentheses indicate that the graph does not include the endpoint  Brackets indicate that the graph does include the endpoint  On a graph, we can see this with open and closed circles  Open Circles indicate we are NOT including the point: ()  Closed Circles indicate that we ARE including the point: [] INTERVAL NOTATION

PRACTICE

 Sometimes we have graphs that increase in more than one place.  Rather than write out the word “and” we use the symbol “  ”  We call this a Union. TRANSLATION:

 Lets check out Pierre the mountain climbing ant! INCREASING/DECREASING

 In this graph, the interval where the parabola is increasing is from -  to -1.  The graph is decreasing from _____ to _____. INCREASING VS. DECREASING

INCREASING VS. DECREASING IN INTERVAL NOTATION

WHAT ARE THE DOMAIN & RANGE IN INTERVAL NOTATION

FIND THE DOMAIN OF THE GRAPHS

 Using our new math vocabulary and our knowledge of interval notation, describe the increasing and decreasing parts of the graph. LET’S LOOK AT THIS GRAPH AGAIN…

 Worksheet- Fill in the table with the information from the picture! HOMEWORK

 Sketch the graph, and identify the following:  Direction of opening  Y-intercept  Vertex  AOS  Zeros (factor)  Max or Min  Increasing and decreasing intervals WARM UPFEB. 11 TH

EQUATIONS PROPERTIES OF QUADRATIC EQUATIONS

Take a look at the graphs of y = x 2 and y = -x 2. THINK/PAIR/SHARE: What is different about the two equations? How does this affect the graph? WE CAN FIND OUT INFORMATION ABOUT A PARABOLA FROM ITS EQUATION

y = Ax 2 + Bx + C STANDARD FORM OF QUADRATIC FUNCTIONS

If a>0, the direction of opening is UP. If a<0, the direction of opening is DOWN. Example: y = 5x x – 7 DIRECTION OF OPENING:

 Y-intercept: in standard form, c gives us the y-intercept. y = ax 2 + bx + c  Example: y = 5x x – 7

 y = -8x 2 + 2x + 1  y = 6x 2 – 24x - 4 GIVE THE DIRECTION OF OPENING AND Y- INTERCEPT OF THE FOLLOWING:

 Axis of symmetry: we can use the formula to find the AOS.  Vertex: plug the AOS in for x and find the y value of the vertex.  Example: y = 5x x – 7

 y = -8x 2 + 2x + 1  y = 6x 2 – 24x - 4 FIND THE AXIS OF SYMMETRY AND VERTEX OF EACH OF THE FOLLOWING:

 Sketch the graph of the parabolas below:  y = -7x x - 2  y = 3x 2 + 4x + 2  What do you predict will happen as the graph continues to the left and right? PREDICTIONS

 If a>0, the end behavior will be that the graph goes “up to the left and up to the right”  If a<0, the end behavior will be that the graph goes “down to the left and down to the right” END BEHAVIOR

 y = x x - 11  y = -x 2 + 8x + 12  y = -2x 2 + 6x + 56  y = 4x 2 - 4x - 32 DESCRIBE THE END BEHAVIOR OF THE GRAPHS WITH THE FOLLOWING EQUATIONS

 The x-values that show where the parabola crosses the x-axis  We will find these values by graphing, factoring, or using the quadratic formula. ZEROS, ROOTS, X-INTERCEPTS

 Graphing: visually identify the intersections.  Factoring: factor the equation then set each set of parentheses equal to 0 and solve for x.  Quadratic Formula: plug A, B and C into the formula and simplify. ZEROS, ROOTS, X-INTERCEPTS

 y = x x - 11  y = -3x  y = -2x 2 + 6x + 56  y = 4x 2 - 8x - 32 FACTOR TO FIND THE ZEROS

HOMEWORK

 Sketch the graph, and identify the following:  Direction of opening  Y-intercept  Vertex  AOS  Zeros (factor)  Max or Min  Increasing and decreasing intervals WARM UPFEB. 12 TH F(x) = -2x 2 + 8x – 3

 You need to make 2 posters today!  You will work TOGETHER with 1-2 other people.  This is not an individual task that multiple names get put on.  I will have up what is needed for every poster.  I will come around and you need to pick 2 pieces of paper, one of each color. (these are the two posters you will make)  This is a QUIZ grade! So use your time wisely and work your hardest! POSTERS

1.Equation 2.Table of x and y values 3.Sketch the graph 4.Find and label the y - intercept 5.Find and label the vertex 6.Give the axis of symmetry 7.Give the min/max value 8.Give the direction of opening 9.Zeros if you can identify them. 10.Interval notation for increasing and decreasing curves ON YOUR POSTER YOU SHOULD INCLUDE THE FOLLOWING: Make sure your names are on the back!

HOMEWORK