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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 SEPT. 11 TH XY -3 -2 0 1 2 3
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September 11 th PROPERTIES OF QUADRATIC EQUATIONS
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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
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y = Ax 2 + Bx + C STANDARD FORM OF QUADRATIC FUNCTIONS
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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?
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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 ALSO FIND OUT INFORMATION ABOUT A PARABOLA FROM ITS EQUATION
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Direction of opening: If a>0, the direction of opening is UP. If a<0, the direction of opening is DOWN. Example: y = 5x 2 + 10x – 7 WE CAN ALSO FIND OUT INFORMATION ABOUT A PARABOLA FROM ITS EQUATION
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Y-intercept: in standard form, c gives us the y-intercept. y = ax 2 + bx + c Example: y = 5x 2 + 10x – 7 WE CAN ALSO FIND OUT INFORMATION ABOUT A PARABOLA FROM ITS EQUATION
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y = -8x 2 + 2x + 1 y = 6x 2 – 24x - 4 GIVE THE DIRECTION OF OPENING AND Y- INTERCEPT OF THE FOLLOWING:
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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?
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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 2 + 10x – 7 WE CAN ALSO FIND OUT INFORMATION ABOUT A PARABOLA FROM ITS EQUATION
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y = -8x 2 + 2x + 1 y = 6x 2 – 24x - 4 FIND THE AXIS OF SYMMETRY AND VERTEX OF EACH OF THE FOLLOWING:
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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:
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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?
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Direction of opening Y-intercept Vertex AOS Max/Min LET’S TAKE A LOOK AT A FEW PARABOLAS
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DIRECTION OF OPENING, Y-INTERCEPT, VERTEX, AOS, MAX/MIN
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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
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What would the graph do if we expanded the view further left and right? We use the parabola’s direction of the opening. END BEHAVIOR
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Sketch the graph of the parabolas below: y = -7x 2 + 15x - 2 y = 3x 2 + 4x + 2 What do you predict will happen as the graph continues to the left and right? PREDICTIONS
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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
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y = x 2 - 10x - 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
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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
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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
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IDENTIFY THE ZEROS IN THE GRAPHS
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y = x 2 - 10x - 11 y = -3x 2 + 12 y = -2x 2 + 6x + 56 y = 4x 2 - 8x - 32 FACTOR TO FIND THE ZEROS
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Worksheet Complete the table You may not be able to find zeros for each row if graph is not clear HOMEWORK
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Sketch the graph, and identify the following: Direction of opening Y-intercept Vertex AOS Zeros (factor) Max or Min WARM UPSEPT. 12 TH f(x) = 2x 2 – 10x + 12
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Sept. 12 th INCREASING, DECREASING, & INTERVAL NOTATION
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The direction of the curve of a parabola Don’t forget about your end behavior!! INCREASING VS. DECREASING
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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
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In this graph, the interval where the parabola is increasing is from - to -1. The graph is decreasing from _____ to _____. INCREASING VS. DECREASING
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Describe where the graphs are increasing and where they are decreasing INCREASING VS. DECREASING
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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
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Use interval notation to describe where the graphs are increasing and decreasing INTERVAL NOTATION
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TRY SOME WITH CIRCLES
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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:
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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…
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Find the interval notation, set notation, and graph on a number line With a partner! INTERVAL NOTATION WORKSHEET
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Worksheet ODDS HOMEWORK
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Sketch the graph, and identify the following: Direction of opening Y-intercept Vertex AOS Zeros (factor) Max or Min Increasing and decreasing intervals WARM UPSEPT. 13 TH F(x) = -2x 2 + 8x – 3
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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. This is a QUIZ grade! So use your time wisely and work your hardest! POSTERS
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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:
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Worksheet from yesterday EVENS HOMEWORK
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