ALEX Classroom Resource

  

Algebra I Module 4, Topic C: Function Transformations and Modeling

  Classroom Resource Information  

Title:

Algebra I Module 4, Topic C: Function Transformations and Modeling

URL:

https://www.engageny.org/resource/algebra-i-module-4-topic-c-overview

Content Source:

EngageNY
Type: Lesson/Unit Plan

Overview:

In Module 4, Topic C, students explore the families of functions that are related to the parent functions, specifically for quadratic (f(x) = x2), square root (f(x) = the square root of x), and cube root (f(x) = cube root of x), to perform horizontal and vertical translations as well as shrinking and stretching (F-IF.C.7b, F-BF.B.3). They recognize the application of transformations in vertex form for a quadratic function and use it to expand their ability to efficiently sketch graphs of square and cube root functions. Students compare quadratic, square root, or cube root functions in context and represent each in different ways (verbally with a description, as a table of values, algebraically, or graphically). In the final two lessons, students examine real-world problems of quadratic relationships presented as a data set, a graph, a written relationship, or an equation. They choose the most useful form for writing the function and apply the techniques learned throughout the module to analyze and solve a given problem (A-CED.A.2), including calculating and interpreting the rate of change for the function over an interval (F-IF.B.6).

Content Standard(s):
Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
12. Create equations in two or more variables to represent relationships between quantities in context; graph equations on coordinate axes with labels and scales and use them to make predictions. Limit to contexts arising from linear, quadratic, exponential, absolute value, and linear piecewise functions.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a contextual situation expressing a relationship between quantities with two or more variables,
  • Model the relationship with equations and graph the relationship on coordinate axes with labels and scales.
  • Make predictions about the contextual situation using the graphs of the equations.
Teacher Vocabulary:
  • Piecewise functions
Knowledge:
Students know:
  • When a particular two variable equation accurately models the situation presented in a contextual problem.
Skills:
Students are able to:
  • Write equations in two variables that accurately model contextual situations.
  • Graph equations involving two variables on coordinate axes with appropriate scales and labels.
  • Make predictions about the context using the graph.
Understanding:
Students understand that:
  • There are relationships among features of a contextual problem, a created mathematical model for that problem, and a graph of that relationship.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.12.1: Solve the equations represented by real-world situations.
ALGI.12.2: Set up an equation to represent the given situation, using correct mathematical operations and variables.
ALGI.12.3: Given a contextual situation, interpret and defend the solution in the context of the original problem.
ALGI.12.4: Explain how to draw informal inferences from data distributions.
ALGI.12.5: Define ordered pair and coordinate plane.
ALGI.12.6: Create equations with two variables (exponential, quadratic and linear).
ALGI.12.7: Graph equations on coordinate axes with labels and scales (exponential, quadratic, and linear).
ALGI.12.8: Identify an ordered pair and plot it on the coordinate plane.

Prior Knowledge Skills:
  • Demonstrate how to plot points on a coordinate plane using ordered pairs from a table.
  • Plot independent (input) and dependent (output) values on a coordinate plane.
  • Draw and label a coordinate plane.
  • Define dependent variable, independent variable, ordered pairs, input, output, and coordinate plane.

Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.11.11 Select an equation or inequality involving one operation (limit to addition or subtraction) with one variable that represents a real-world problem. Solve the equation.


Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
23. Identify the effect on the graph of replacing f(x) by f(x)+k,k·f(x), f(k·x), 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 explain the effects on the graph, using technology as appropriate. Limit to linear, quadratic, exponential, absolute value, and linear piecewise functions.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a function in algebraic form,
  • Graph the function, f(x), conjecture how the graph of f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k(both positive and negative) will change from f(x), and test the conjectures.
  • Describe how the graphs of the functions were affected (e.g., horizontal and vertical shifts, horizontal and vertical stretches, or reflections).
  • Use technology to explain possible effects on the graph from adding or multiplying the input or output of a function by a constant value.

  • Given the graph of a function and the graph of a translation, stretch, or reflection of that function, determine the value which was used to shift, stretch, or reflect the graph.
Teacher Vocabulary:
  • Composite functions
  • Horizontal and vertical shifts
  • Horizontal and vertical stretch
  • Reflections
  • Translations
Knowledge:
Students know:
  • Graphing techniques of functions.
  • Methods of using technology to graph functions
Skills:
Students are able to:
  • Accurately graph functions.
  • Check conjectures about how a parameter change in a function changes the graph and critique the reasoning of others about such shifts.
  • Identify shifts, stretches, or reflections between graphs.
Understanding:
Students understand that:
  • Graphs of functions may be shifted, stretched, or reflected by adding or multiplying the input or output of a function by a constant value.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.23.1: Define dilation, rotation, reflection, translation, congruent and sequence.
ALGI.23.2: Identify rotations.
ALGI.23.3: Identify reflections.
ALGI.23.4: Identify translations.
ALGI.23.5: Use digital tools to formulate solutions to authentic problems (Ex: electronic graphing tools, probes, spreadsheets).

Prior Knowledge Skills:
  • Identify congruent figures.
  • Compare rotations to translations.
  • Compare reflections to rotations.
  • Compare translations to reflections.
  • Recognize translations (slides), rotations (turns), and reflections (flips).

Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.12.24 Given a simple linear function on a graph, select the model that represents an increase by equal amounts over equal intervals.


Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
29. 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. Limit to linear, quadratic, exponential, and absolute value functions.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Given an interval on a graph or table, calculate the average rate of change within the interval.
  • Given a graph of contextual situation, estimate the rate of change between intervals that are appropriate for the summary of the context.
Teacher Vocabulary:
  • Average rate of change
  • Intervals
Knowledge:
Students know:
  • Techniques for graphing.
  • Techniques for finding a rate of change over an interval on a table or graph.
  • Techniques for estimating a rate of change over an interval on a graph.
Skills:
Students are able to:
  • Calculate rate of change over an interval in a table or graph.
  • Estimate a rate of change over an interval on a graph.
Understanding:
Students understand that:
  • The average provides information on the overall changes within an interval, not the details within the interval (an average of the endpoints of an interval does not tell you the significant features within the interval).
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.29.1: Identify equivalent ratios.
ALGI.29.2: Define average rate of change as slope.
ALGI.29.3: Estimate the rate of change from a graph (rise/run).
ALGI.29.4: Interpret the average rate of change.
ALGI.29.5: Calculate the average rate of change.
ALGI.29.6: Compute the slope of a line given two ordered pairs.
ALGI.29.7: Identify the slope, given slope-intercept form.

Prior Knowledge Skills:
  • Apply the identification of the slope and the y-intercept to a real-world situation.
  • Recall how to write an equation in slope-intercept form.
  • Recall how to solve equations by using substitution.
  • Identify how many solutions the linear equation may or may not have.
  • Calculate an expression in the correct order (Ex. exponents, mult./div. from left to right, and add/sub. from left to right).
  • Define ratio, rate, proportion, percent, equivalent, input, output, ordered pairs, diagram, unit rate, and table.

Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.12.28 Given graphs that represent linear functions, identify key features (limit to y intercept, x-intercept, increasing, decreasing) and/or interpret different rates of change (e.g., Which is faster or slower?).


Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
30. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

b. Graph piecewise-defined functions, including step functions and absolute value functions.

c. Graph exponential functions, showing intercepts and end behavior.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a symbolic representation of a function (including linear, quadratic, absolute value, piecewise-defined functions, and exponential,
  • Produce an accurate graph (by hand in simple cases and by technology in more complicated cases) and justify that the graph is an alternate representation of the symbolic function.
  • Identify key features of the graph and connect these graphical features to the symbolic function, specifically for special functions:
      quadratic or linear (intercepts, maxima, and minima) and piecewise-defined functions, including step functions and absolute value functions (descriptive features such as the values that are in the range of the function and those that are not).
    1. Exponential (intercepts and end behavior).
Teacher Vocabulary:
  • x-intercept
  • y-intercept
  • Maximum
  • Minimum
  • End behavior
  • Linear function
  • Factorization
  • Quadratic function
  • Intercepts
  • Piece-wise function
  • Step function
  • Absolute value function
  • Exponential function
  • Domain
  • Range
  • Period
  • Midline
  • Amplitude
  • Zeros
Knowledge:
Students know:
  • Techniques for graphing.
  • Key features of graphs of functions.
Skills:
Students are able to:
  • Identify the type of function from the symbolic representation.
  • Manipulate expressions to reveal important features for identification in the function.
  • Accurately graph any relationship.
Understanding:
Students understand that:
  • Key features are different depending on the function.
  • Identifying key features of functions aid in graphing and interpreting the function.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.30.1: Define piecewise-defined functions and step functions.
ALGI.30.2: Graph functions expressed symbolically by hand in simple cases.
ALGI.30.3: Graph functions expressed symbolically using technology for a more complicated case.

a.
ALGI.30.4: Graph quadratic functions showing maxima and minima.
ALGI.30.5: Graph quadratic functions showing intercepts.
ALGI.30.6: Graph linear functions showing intercepts.

b.
ALGI.30.7: Define square root, cube root, and absolute value function.
ALGI.30.8: Graph piecewise-defined functions.
ALGI.30.9: Graph step functions.
ALGI.30.10: Graph cube root functions.
ALGI.30.11: Graph square root functions.
ALGI.30.12: Graph absolute value functions.

c.
ALGI.30.13 Identify exponential numbers as repeated multiplication.
ALGI.30.14 Rewrite exponential numbers as repeated multiplication.

Prior Knowledge Skills:
  • Demonstrate how to plot points on a coordinate plane using ordered pairs from a table.
  • Graph a function given the slope-intercept form of an equation.
  • Recognize the absolute value of a rational number is its' distance from 0 on a vertical and horizontal number line.
  • Define absolute value and rational numbers.
  • Recall how to plot ordered pairs on a coordinate plane.
  • Name the pairs of integers and/or rational numbers of a point on a coordinate plane.

Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.12.30 Given the graph of a linear function, identify the intercepts, the maxima, and minima.


Tags: Algebraic expressions, cube root, equations, graph, intercepts, linear functions, piecewise defined, quadratic functions, rate of change, square root, variables
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There are seven lessons on this topic.

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Author: Hannah Bradley