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Course 1, Unit 3 - Linear Functions

Overview
In the Linear Functions unit, students learn how to recognize situations in which key variables change at a constant rate. They learn how to express and interpret those patterns of change in data tables, straight line graphs, and rules in the form y = a + bx. They identify and interpret the slopes and intercepts of linear graphs. They learn techniques for solving linear equations and inequalities that arise in science and business problems. They also learn to evaluate and solve linear equations using graphs, tables, paper-and-pencil techniques.

Students will frequently use the language "rate of change" when talking about the "slope" of a linear function. (See page 155.) Also note that the letter "b" is not reserved almost exclusively for the y-intercept as you may have experienced in your mathematics program. The y = ax + b form emphasizes the modeling or statistical approach to linear functions where a is the starting value or y-intercept and b is the rate of change. One goal is to help students develop flexible use of variables. Thus, you will notice many different letters representing linear functions throughout the curriculum.

Further work toward developing proficiency with manipulating symbols occurs in subsequent units and courses. Practice for the skills developed in this unit is incorporated in the On Your Own Review tasks and the Practicing for Standardized Test masters at the end of this and subsequent units.

Key Ideas from Course 1, Unit 3

  • Linear data patterns: A relationship between variables x and y is called linear if the graph of related (x, y) values is a straight line. This graph pattern occurs when there is a constant difference between successive y values as x values change uniformly. That is, the ratio (change in y)/(change in x) is constant. For example, in the next graph and accompanying table, (5.5 - 1)/(5 - 2) = (13 - 7)/(10 - 6) = 1.5.

    x
    2
    5
    6
    10
    y
    1
    5.5
    7
    13
  • Linear function rules (equations): Every linear relationship can be expressed with an algebraic rule in the form y = a + bx, where a indicates the y-intercept, (0, a) of the graph and b indicates the slope of the graph and the rate of change in y values. For example, the graph of y = 5 + 3x crosses the y-axis at (0, 5) and rises at a slope of 3 - for every increase of 1 in the value of x, the value of y increases by 3. Because linear functions are those with constant rates of change in the dependent variable, they can also be thought about and represented with NOW-NEXT equations. The linear relation with rule y = 5 + 3x is the same as that with rule NEXT = NOW + 3, starting at 5. (See pages 157-161.)

  • Rate of change: Appears in the table as above, on the graph as slope, in the equation as the coefficient of x. (For example, in y = 5 + 3x, the rate of change is 3.)

  • Modeling linear data patterns: Exact linear relationships can be expressed with algebraic rules in the form y = a + bx. But in many practical problems, experimental data might suggest, but not exactly fit, a linear pattern. In such cases, it is often useful to summarize the data trend by drawing a line that matches the scatterplot of (x, y) data pairs and to find the rule (formula) relating y- and x-coordinates of points on that line. For example, the following graph shows 7 data points and the graph of y = -0.5 + x that matches the pattern in those data quite well.

  • Solving a linear equation: Students have graphic, numeric, and symbolic strategies for finding the value of x that makes equations like a + bx = c true. For example, to solve the equation 5 + 4x = 13, they can scan a table or graph of y = 5 + 4x in search of points with coordinates (x, 13). They can also apply properties of equality to reason like this:
  • If 5 + 4x = 13, then 4x = 8 (subtract 5 from both sides).
    If 4x = 8, then x= 2 (divide both sides by 4).

  • Solving systems of linear equations: In this unit, students solve simple systems of linear equations like y = 3x and y = 5 - 2x. That means finding a pair of values (x, y) that satisfy both conditions. Once again, they have at least three strategies available for this kind of problem. They can graph the two linear functions and look for coordinates of the intersection point. They can scan tables of values for the two functions, looking for points where a single x value produces the same y value for each.

    X Y1 Y2
    0 0 5
    0.5 1.5 4
    1 3 3
    1.5 4.5 2
    2 6 1

    They can also use symbolic reasoning as follows:

    If y = 3x and y = 5 - 2x, then 3x = 5 - 2x.
    So, 5x = 5 (add 2x to both sides).
    So, x = 1 (divide both sides by 5).
    So, y = 3 (substitute x = 1 in either function rule).

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