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Mathematics > Grade 8 > Functions
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Mathematics | Grade : 8

Domain - Functions

Cluster - Define, evaluate, and compare functions.

[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.

Resources:


  • Function
    A mathematical relation for which each element of the domain corresponds to exactly one element of the range.
  • Linear function
    A function with an equation of the form y = mx + b, where m and b are constants

Predecessor Standards:

No Predecessor Standards found.

Successor Standards:

  • 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-REI.D.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 and make tables of values. Include cases where f(x) and/or g(x) are linear and exponential functions.*
  • AI.F-IF.A.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 (range) of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
  • 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-BF.B.3
    Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Include linear, quadratic, exponential, and absolute value functions. Utilize technology to experiment with cases and illustrate an explanation of the effects on the graph.
  • AI.F-LE.A.1
    Distinguish between situations that can be modeled with linear functions and with exponential functions.*
  • 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.1.b
    Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.*
  • AI.F-LE.A.1.c
    Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.*
  • MI.A-REI.D.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 and/or make tables of values. Include cases where f(x) and/or g(x) are linear and exponential functions.*
  • MI.F-IF.A.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).
  • MI.F-BF.B.3
    Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Include linear and exponential models. (Focus on vertical translations for exponential functions). Utilize technology to experiment with cases and illustrate an explanation of the effects on the graph.
  • MI.F-LE.A.1
    Distinguish between situations that can be modeled with linear functions and with exponential functions.*
  • 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.1.b
    Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.*
  • MI.F-LE.A.1.c
    Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.*
  • 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.

Same Level Standards:

  • 8.EE.B.6
    Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
  • 8.F.A.1
    Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. [Note: Function notation is not required in grade 8.]
  • 8.F.A.2
    Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
  • 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.
  • 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.