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Algebra II Honors—Day 10

In today's Algebra II Honors class, students will engage in group investigations focused on the absolute value function. The session will highlight how the parameters a, b, c, and d in the function f(x) = a|bx + c| + d affect its graph. Students will collaboratively explore and graph the parent function, f(x) = |x|, and analyze the impacts of transformations. While preparing for the first graded homework assignment due tomorrow, students will work without calculators to develop their understanding of absolute value graphs.

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Algebra II Honors—Day 10

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  1. Algebra II Honors—Day 10

  2. Procedures • Pick up the following from the table: • Handout, whiteboard, marker, eraser • Get into groups of three or four students.

  3. Goals for Today • Reminder—First Graded Homework Assignment (checked for accuracy)—tomorrow—Tuesday, Sept. 10 • Quotable Puzzle due today • No homework check today • Essential Questions • New Material/Group Investigations • Homework

  4. Essential Questions • How do the values of a, b, c, and d in the function affect the graph of ?

  5. The absolute value function is a function of the form It has a two-part definition as follows: for and for Exploring the Absolute Value Function

  6. The Graph For x<0, the graph is the same as the line y=-x (intercept of 0 and slope of NEGATIVE 1) For x≥0, the graph is the same as the line y=x (intercept of 0 and slope of POSITIVE 1) y=|x|

  7. Explanations • f(x)=a|bx+c|+d The letters a, b, c, and d represent shifts or changes to the basic “parent” graph of f(x)=|x|. Each of these plays a different role in the movement of the graph. You will explore each.

  8. Assignment • Within each group divide up the work in each section on the handout and then compare answers for all the parts. Use the whiteboards first until all graphs are finished. • Draw the basic graph: f(x)=|x|. This is called the “parent graph” or the “parent function.” • Make a table of points for the other equations and graph each one on the same coordinate plane. For each graph, use the domain {-3, -1, 0, 1, 3} and find the y-values. Then draw the graph.

  9. The effect of “a” (the number outside the function—multiplies the entire function) The parent function • New function “grows” faster • called a “vertical stretch” • slope is steeper—multiplied by “a” • New function “grows” slower • called a “vertical shrink” • slope is less steep—multiplied by “a” • the negative flips it upside down as well

  10. The effects of “b” and “c” (the numbers inside the function—“b” multiplies the x only) The parent function • b=1 and c=3 • New function is shifted LEFT 3 units (“-c”) and slope is still the same as parent • b=1 and c=-5 • New function is shifted RIGHT 5 units (“-c”) and slope is still the same as parent

  11. The effects of “b” and “c” (the numbers inside the function—“b” multiplies the x only) The parent function • b=6 and c=3 • New function is shifted LEFT 1/2 unit (“-c/b”) and slope is multiplied by 6 (“b”) • b=3 and c=-9 • New function is shifted RIGHT 3 units (“-c/b”)and slope is multiplied by 3 (“b”)

  12. The effects of “d” (the number added outside the function) The parent function • d=8 • New function is shifted UP 8 units (“d”) • d=-6 • New function is shifted DOWN 6 units (“d”)

  13. In your notes • Make sure you can summarize the effects of each of the numbers a, b, c, and d in the equation . These numbers will be used throughout this course for other functions. • Understand that each of these numbers either “shifts” or “stretches” the parent function.

  14. Homework • Absolute Value Graphs • Do without a calculator based on what you learned. Problems 1, 2, 4, 7, 8, 9, 10, 12

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