Unit 3: Linear and Exponential Functions

[Pages:59]Coordinate Algebra EOCT

UNIT 3: LINEAR AND EXPONENTIAL FUNCTIONS

Unit 3: Linear and Exponential Functions

In Unit 3, students will learn function notation and develop the concepts of domain and range. They will discover that functions can be combined in ways similar to quantities, such as adding. Students will explore different ways of representing functions (e.g., graphs, rules, tables, and sequences) and interpret functions given graphically, numerically, symbolically, and verbally. Discovering how functions can be transformed, similar to shapes in geometry, and learning about how parameters affect functions are aspects of this unit. Students will also learn how to compare and contrast linear and exponential functions, distinguishing between additive and multiplicative change. They will interpret arithmetic sequences as linear functions and geometric sequences as exponential functions.

KEY STANDARDS

Represent and solve equations and inequalities graphically

MCC9-12.A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). (Focus

on linear and exponential equations and be able to adapt and apply that learning to other types of equations in future courses.)

MCC9-12.A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

Understand the concept of a function and use function notation

MCC9-12.F.IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). (Draw examples from linear and exponential functions.)

MCC9-12.F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. (Draw examples from linear and exponential functions.)

MCC9-12.F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n 1 (n is greater than or equal to 1). (Draw connection to F.BF.2, which requires students to write arithmetic and geometric sequences.)

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Coordinate Algebra EOCT

UNIT 3: LINEAR AND EXPONENTIAL FUNCTIONS

Interpret functions that arise in applications in terms of the context

MCC9-12.F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. (Focus on linear and exponential functions.)

MCC9-12.F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. (Focus on linear and exponential functions.)

MCC9-12.F.IF.6 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. (Focus on linear functions and intervals for exponential functions whose domain is a subset of the integers.)

Analyze functions using different representations

MCC9-12.F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. (Focus on linear and exponential functions. Include comparisons of two functions presented algebraically.)

MCC9-12.F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima.

MCC9-12.F.IF.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

MCC9-12.F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. (Focus on linear and exponential functions. Include comparisons of two functions presented algebraically.)

Build a function that models a relationship between two quantities

MCC9-12.F.BF.1 Write a function that describes a relationship between two quantities. (Limit to linear and exponential functions.)

MCC9-12.F.BF.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. (Limit to linear and exponential functions.)

MCC9-12.F.BF.1b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant

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Coordinate Algebra EOCT

UNIT 3: LINEAR AND EXPONENTIAL FUNCTIONS

function to a decaying exponential, and relate these functions to the model. (Limit to linear and exponential functions.)

MCC9-12.F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

Build new functions from existing functions MCC9-12.F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(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. (Focus on vertical translations of graphs of linear and exponential functions. Relate the vertical translation of a linear function to its y-intercept.)

Construct and compare linear, quadratic, and exponential models and solve problems MCC9-12.F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.

MCC9-12.F.LE.1a Prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.

MCC9-12.F.LE.1b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

MCC9-12.F.LE.1c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

MCC9-12.F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

MCC9-12.F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Interpret expressions for functions in terms of the situation they model MCC9-12.F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context. (Limit exponential functions to those of the form f(x) = bx + k.)

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Coordinate Algebra EOCT

UNIT 3: LINEAR AND EXPONENTIAL FUNCTIONS

REPRESENT AND SOLVE EQUATIONS AND INEQUALITIES GRAPHICALLY

KEY IDEAS

1. The graph of a linear equation in two variables is a collection of ordered pair solutions in a

coordinate plane. It is a graph of a straight line. Often tables of values are used to organize

the ordered pairs.

Example:

Every year Silas buys fudge at the state fair. He buys peanut butter and chocolate. This year he intends to buy $24 worth of fudge. If chocolate costs $4 per pound and peanut butter costs $3 per pound, what are the different combinations of fudge that he can purchase if he only buys whole pounds of fudge?

Solution:

If we let x be the number of pounds of chocolate and y be the number pounds of peanut butter, we can use the equation 4x + 3y = 24. Now we can solve this equation for y to make it easier to complete our table.

4x + 3y = 24

Write the original equation.

4x ? 4x + 3y = 24 ? 4x

Subtract 4x from each side.

3y = 24 ? 4x

Simplify.

3y 3

24 3

4x

Divide each side by 3.

y

24

3

4x

Simplify.

We will only use whole numbers in the table, because Silas will only buy whole pounds of the fudge.

Chocolate,

x

0 1 2 3 4 5 6

Peanut butter,

y

8 6 (not a whole number) 5 (not a whole number)

4 2 (not a whole number) 1 (not a whole number)

0

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Coordinate Algebra EOCT

UNIT 3: LINEAR AND EXPONENTIAL FUNCTIONS

The ordered pairs from the table that we want to use are (0, 8), (3, 4), and (6, 0). The graph would look like the one shown below:

Based on the number of points in the graph, there are three possible ways that Silas can buy pounds of the fudge: 8 pounds of peanut butter only, 3 pounds of chocolate and 4 pounds of peanut butter, or 6 pounds of chocolate only. Notice that if the points on the graph were joined, they would form a line. If Silas allowed himself to buy partial pounds of fudge, then there would be many more possible combinations. Each combination would total $24 and be represented by a point on the line that contains (0, 8), (3, 4), and (6, 0).

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Coordinate Algebra EOCT

UNIT 3: LINEAR AND EXPONENTIAL FUNCTIONS

Example:

Silas decides he does not have to spend exactly $24 on the fudge, but he will not spend more than $24. What are the different combinations of fudge purchases he can make?

Solution:

Our new relationship is the inequality 4x + 3y 24. We'll make a new table of values. We begin by making it easier to find the number of pounds of peanut butter by solving for y.

4x 3y 24

Write the original equation.

4x 4x 3y 24 4x

Subtract 4x from each side.

3y 24 4x

Simplify.

3y 3

24 4x 3

Divide each side by 3.

y

24

3

4 x

Simplify.

Once again, we will only use whole numbers from the table, because Silas will only buy whole pounds of the fudge.

Chocolate, x

0 1 2 3 4 5 6

Peanut butter,

y

?

24

3

4

x

0, 1, 2, 3, 4, 5, 6, 7, 8

0, 1, 2, 3, 4, 5, 6, 6

0, 1, 2, 3, 4, 5, 5

0, 1, 2, 3, 4

0, 1, 2, 2

0, 1, 1

0

Each row in the table has at least one value that is a whole number in both columns. If Silas does not intend to spend all of his $24, there are many more combinations. Let's look at the graph.

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Coordinate Algebra EOCT

UNIT 3: LINEAR AND EXPONENTIAL FUNCTIONS

From the points that lie on the graph and the rows in the table, there are 33 combinations of chocolate and peanut butter fudge that Silas can buy. The points on the graph do not look like a line. However, they all appear to fall on or below the line that would join (0, 8), (3, 4), and (6, 0). In fact, the line joining those three points would be the boundary line for the inequality that represents the combinations of pounds of fudge that Silas can purchase. The points do not cover the entire half-plane below that line because it is impossible to purchase negative pounds of fudge. The fact that you cannot purchase negative amounts of fudge puts constraints, or limitations, on the possible x and y values. Both x and y values must be non-negative.

REVIEW EXAMPLES

1) Consider the equations y = 2x ? 3 and y = ?x + 6.

a. Complete the tables below, and then graph the equations on the same coordinate axes.

y = 2x-3

y =-x +6

x

y

x

y

b. Is there an ordered pair that satisfies both equations? If so, what is it?

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Coordinate Algebra EOCT

UNIT 3: LINEAR AND EXPONENTIAL FUNCTIONS

c. Graph both equations on the same coordinate plane by plotting the ordered pairs from the tables and connecting the points.

d. Do the lines appear to intersect? If so, where? How can you tell that the point where the lines appear to intersect is a common point for both lines?

Solution:

a.

y = 2x ? 3

y = ?x + 6

x

y

x

y

?1

?5

?1

7

0

?3

0

6

1

?1

1

5

2

1

2

4

3

3

3

3

b. Yes, the ordered pair (3, 3) satisfies both equations.

c. d. The lines appear to intersect at (3, 3). When x = 3 and y = 3 are substituted into each

equation, the values satisfy both equations. This proves that (3, 3) lies on both lines, which means it is a common solution to both equations.

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