Content Area: Math

 

Index: 4.3A Grade 12 CPI 3

 

Standard: 4.3 - Patterns and Algebra

 

Strand: A - Patterns

 

Cumulative Progress Indicator: 3 - The student will understand and perform transformations on commonly-used functions

·        Translations, reflections, dilations

·        Effects on linear and quadratic graphs of parameter changes in equations

·        Using graphing calculators or computers for more complex functions

 

Grade: 12

 

Sample Activities:

 

·        Students investigate the characteristics of linear functions. For example, in y = kx, how does a change in k affect the graph? In y = mx + b, what is the role of b? Does k in the first equation serve the same purpose as m in the second? Students use the graphing calculator to investigate and verify their conclusions.

 

·        Students investigate the effects of a dilation and/or a horizontal or vertical shift on the algebraic expression of various types of functions. For example, how does moving a graph up 3 units affect its equation?

 

·        Students look at the effects of changing the coefficients of a quadratic equation on its graph. For example, how is the graph of y = 4x^2 different from that of y = x^2? How is y = .2x^2 different from y = x^2? How are y = x^2 + 4, y = x^2 - 4, y = x^2 - 4x, and y = x^2 - 4x + 4 each different from y = x^2? How is y = sin 4x different from y = 4 sin x? Students use graphing calculators to look at the graphs and summarize their conjectures in writing.

 

·        Students work in groups to investigate what size square to cut from each corner of a rectangular piece of cardboard in order to make the largest possible open-top box. They make models, record the size of the square and the volume for each model, and plot the points on a graph. They note that the relationship seems to be a polynomial function and make a conjecture about the maximum volume, based on the graph. The students also generate a symbolic expression describing this situation and check to see if it matches their data by using a graphing calculator.

 

·        Students look at the effects of changing the coefficients of a trigonometric equation on the graph. For example, how is the graph of y = 4 sin x different from that of y = sin x? How is y = .2 sin x different from y = sin x? How are y = sin x + 4, y = sin x 4, y = sin (x 4), and y = sin (x + 4) each different from y = sin x? Students use graphing calculators to look at the graphs and summarize their conjectures in writing.

 

·        Students begin with the graph of y = 2x. They shift the graph up one unit and try to find the equation of the resulting curve. They shift the original graph one unit to the right and try to find the equation of that curve. They reflect the original graph across the x-axis and try to find the equation of that curve. Finally, they reflect the original graph across the y-axis and try to find the equation of the resulting curve. They describe what they have learned in their journals.