Plot each point on graph paper.You will attach graph paper to Bell Ringer sheet. 1. A(0,0) 2. B(5,0) 3. C(–5,0) 4. D(0,5) 5. E(0, –5) A A B CC D D E E Bell Ringer Domain ? Range ? Intercepts
Holt McDougal Common Core Edition 1.1 1.4 F-IF.6 A-REI.1 F-BF.3
Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.*
Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx) and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards. The basic modeling cycle involves (1) identifying variables in the situation and selecting those that represent essential features, (2) formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables, (3) analyzing and performing operations on these relationships to draw conclusions, (4) interpreting the results of the mathematics in terms of the original situation, (5) validating the conclusions by comparing them with the situation, and then either improving the model or, if it is acceptable, (6) reporting on the conclusions and the reasoning behind them. Choices, assumptions, and approximations are present throughout this cycle.
Correlation Slope Reflection Regression Stretch Transformation vs. Translation Parent Function Due: 28 th, Test 1 Date Copy words for Bell Ringer on Bell Ringer sheet
teacher note Parent Function o Definition: the simplest function with the defining characteristics of the family. o Characteristics: functions in the same family are transformations of their parent function. o Example: f(x)=x^2 is parent function of f(x)=x^ o Picture: Transformation o Definition: a change in the position, size, or shape of a figure or graph. o Characteristics: same shape, increase or decrease in size or location. o Example: o Picture: Translation o Definition: slide; same figure and size just moved. o Characteristics: identical o Example: o Picture: (1,2)(2,3) Point out word correlations: trans sl ation Sl ide
teaching note students need calculators students need graph paper On next slide allow time to draw “Slope Tree” if needed
Slope tree positive negative undefined
Perform the given translation on the point (–3, 4). Give the coordinates of the translated point. Example: Translating Points 5 units right Translating (–3, 4) 5 units right results in the point (2, 4). (2, 4) (-3, 4) ? Slope ? ? Domain ? ? Range ?
2 units left and 3 units down Translating (–3, 4) 2 units left and 3 units down results in the point (–5, 1). (–3, 4) (–5, 1) 2 units 3 units Perform the given translation on the point (–3, 4). Give the coordinates of the translated point. Example: Translating Points
Example: Translation 4 units right Perform the given translation on the point (–1, 3). Give the coordinates of the translated point. Translating (–1, 3) 4 units right results in the point (3, 3). (–1, 3) (3, 3) ? Slope ?
Example: Translation 1 unit left and 2 units down Perform the given translation on the point (–1, 3). Give the coordinates of the translated point. Translating (–1, 3) 1 unit left and 2 units down results in the point (–2, 1). (–1, 3) (–2, 1) 1 unit 2 units ? Slope ?
Notice that when you translate left or right, the x-coordinate changes, and when you translate up or down, the y-coordinate changes. Translations Horizontal TranslationVertical Translation
Translations Horizontal TranslationVertical Translation Bell Ringer Using complete sentence, correlate the “h” and “k” to “x” and “y”.
Teaching note have student in each class add congruent to vocabulary on walls after next slide
You can transform a function by transforming its ordered pairs. When a function is translated or reflected, the original graph and the graph of the transformation are congruent because the size and shape of the graphs are the same. vocabulary ACT
teaching instruction Can I do this on a calculator? Graph y=x on a graphing calculator by pressing y= and entering y1=x. Then press graph. Enter and graph y2 = x + 5. Describe the graph of y2 as compared to y1. Enter and graph y3= x – 5. describe the graph of y3 as compared to y1. save as Exit Question, tell students to think about it: Make a conjecture about how a change in the value of k in the equation y=x + k affects the equation’s graph.
Start 4 page packet You need to be responsible and keep up with it We will work on each day It is not homework yet
Activity Work with a partner (optional) and think of the classroom as a large coordinate plane. Consider your seating arrangement in class. Identify your seat. Then describe where you would like to sit. Use the aisles and rows as coordinates. Write down your steps, process and justify your reasoning. You must move a minimum of 7 spaces on each axis. Work with a partner: both names on same page of work
You graphed y=x on a graphing calculator by pressing y= and entering y1=x. Then pressed graph. Then you entered and graphed y2 = x + 5. Next you entered and graphed y3= x – 5. How does the term congruent explain how a change in the value of k in the equation Exit Question: How does the term congruent explain how a change in the value of k in the equation y=x + k affects the equation’s graph.
On graph paper. You may use the same piece from yesterday’s Bell Ringer. Perform the given translation on the point (-3, 4). Indicate the coordinates of the translated point. This is two different translations. a) 5 units right b) 2 unit left and 2 units down
teaching note: following slide ask students what they can summarize about reflections. – If it flips across Y-axis, the X changes – If it flips across X-axis, the Y changes – The sign changes; point out it does not always become negative; only negative if starts positive.
Reflections Reflection Across y-axisReflection Across x-axis
teaching note following slide *recommend students get out colored pencils; *ask student what are “important” points on a graph – intercepts – end points, to keep same shape *ask students, what are we changing, the X or Y? and by how much? -y and adding 2 (up) * at end of next slide, point out X does not change **red orange orange green green blue blue purple
Example translation 2 units up Identify important points from the graph and make a table. The entire graph shifts 2 units up. Add 2 to each y-coordinate = = = = = 2 X Y Y + 2 ? (green) Slope(s) ? Team of no more than 2; winning group gets prize! Up is K value K is your Y
Example translation 3 units right Use a table to perform the transformation of y = f(x). Use the same coordinate plane as the original function. The entire graph shifts 3 units right. Add 3 to each x-coordinate. xyx + 3 –24–2 + 3 = 1 –10–1 + 3 = = 3 1)What axis are you changing 2)Id key points on original graph 3)Plot new ordered pairs steps What did you do? What happened?
teaching note following slide show students following slide without revealing bottom ask students what happened – then reveal
reflection across x-axis xy–y –24–4 –100 02–2 22 f Multiply each y-coordinate by –1. The entire graph flips across the x-axis. Example Use a table to perform the transformation of y = f(x). Use the same coordinate plane as the original function. Steps ??
Example Business Application The graph shows the cost of painting based on the number of cans of paint used. Sketch a graph to represent the cost of a can of paint doubling, and identify the transformation of the original graph that it represents. If the cost of painting is based on the number of cans of paint used and the cost of a can of paint doubles, the cost of painting also doubles. This represents a vertical stretch by a factor of 2. Read only
Perform each transformation of y = f(x). Use the same coordinate plane as the original function. Copy the original function. Create a table. You can do all on graph paper a) Translate 2 units up b) Reflection across x axis