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Mathematics > Grade 8 > Functions
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Mathematics | Grade : 8

Domain - Functions

Cluster - Use functions to model relationships between quantities.

[8.F.B.4] - Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Resources:

Predecessor Standards:

  • 7.RP.A.2
    Recognize and represent proportional relationships between quantities.

Successor Standards:

  • AI.A-SSE.A.1.a
    Interpret parts of an expression, such as terms, factors, and coefficients.
  • AI.A-SSE.A.1.b
    Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)t as the product of P and a factor not depending on P.
  • AI.A-SSE.B.3
    Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
  • AI.A-SSE.B.3.a
    Factor a quadratic expression to reveal the zeros of the function it defines.
  • AI.A-SSE.B.3.b
    Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
  • AI.A-SSE.B.3.c
    Use the properties of exponents to transform expressions for exponential functions. For example, the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
  • AI.A-CED.A.1
    Create equations and inequalities in one variable and use them to solve problems. (Include equations arising from linear, quadratic, and exponential functions with integer exponents.)*
  • AI.A-CED.A.2
    Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.*
  • AI.F-IF.B.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.*
  • AI.F-IF.C.8.a
    Use the process of factoring and completing the square in a quadratic function to show zeros, maximum/minimum values, and symmetry of the graph, and interpret these in terms of a context.
  • AI.F-IF.C.8.b
    Use the properties of exponents to interpret expressions for exponential functions. Apply to financial situations such as identifying appreciation and depreciation rate for the value of a house or car some time after its initial purchase: Vn=P(1+r)n. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, and y = (1.2)t /10, and classify them as representing exponential growth or decay.
  • AI.F-BF.A.1.a
    Determine an explicit expression, a recursive process, or steps for calculation from a contex.*
  • AI.F-BF.A.1.b
    Combine standard function types using arithmetic operations.* For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
  • AI.F-BF.A.2
    Write arithmetic and geometric sequences both recursively and with an explicit formula them to model situations, and translate between the two forms.*
  • AI.F-LE.A.1.a
    Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.*
  • AI.F-LE.A.2
    Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (including reading these from a table).*
  • AI.F-LE.B.5
    Interpret the parameters in a linear or exponential function (of the form f(x) = bx + k) in terms of a context.*
  • AI.S-ID.C.7
    Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.*
  • MI.A-SSE.A.1.a
    Interpret parts of an expression, such as terms, factors, and coefficients.
  • MI.A-SSE.A.1.b
    Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P.
  • MI.A-CED.A.1
    Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and exponential functions with integer exponents.*
  • MI.A-CED.A.2
    Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.*
  • MI.F-IF.B.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.*
  • MI.F-BF.A.1.a
    Determine an explicit expression, a recursive process, or steps for calculation from a context.*
  • MI.F-BF.A.1.b
    Combine standard function types using arithmetic operations.* For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
  • MI.F-BF.A.2
    Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.*
  • MI.F-LE.A.1.a
    Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.*
  • MI.F-LE.A.2
    Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (including reading these from a table).*
  • MI.F-LE.B.5
    Interpret the parameters in a linear function or exponential function (of the form f(x) = bx + k) in terms of a context.*
  • MI.S-ID.C.7
    Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.*
  • MII.A-SSE.B.3
    Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
  • MII.A-SSE.B.3.b
    Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
  • MII.A-SSE.B.3.c
    Use the properties of exponents to transform expressions for exponential functions. For example, the expression 1.15t can be rewritten as (1.15 1/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
  • MII.F-IF.C.8.a
    Use the process of factoring and completing the square in a quadratic function to show zeros, minimum/maximum values, and symmetry of the graph and interpret these in terms of a context.
  • MII.F-IF.C.8.b
    Use the properties of exponents to interpret expressions for exponential functions. Apply to financial situations such as Identifying appreciation/depreciation rate for the value of a house or car some time after its initial purchase: Vn=P(1+r)n. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, and y = (1.2) t /10, and classify them as representing exponential growth or decay.

Same Level Standards:

  • 8.F.A.3
    Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
  • 8.F.B.5
    Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
  • 8.SP.A.2
    Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
  • 8.SP.A.3
    Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.