Math Problem Statement

Identifying The Characteristics of Linear and Exponential Functions Function Type and Behavior d(y)=6(4)y Select an answer Exponential Linear Select an answer Increasing Decreasing p(t)=60(1.2)t Select an answer Exponential Linear Select an answer Increasing Decreasing g(x)=-14x+130 Select an answer Exponential Linear Select an answer Increasing Decreasing h(x)=220(0.9)x Select an answer Exponential Linear Select an answer Increasing Decreasing f(x)=28x-50 Select an answer Exponential Linear Select an answer Increasing Decreasing

Solution

Let's analyze the given functions one by one to identify their type (linear or exponential) and behavior (increasing or decreasing):

  1. d(y)=6(4)yd(y) = 6(4)^y

    • Type: This is an exponential function because the variable yy is in the exponent.
    • Behavior: Since the base 4>14 > 1, the function is increasing.
      Answer: Exponential, Increasing.
  2. p(t)=60(1.2)tp(t) = 60(1.2)^t

    • Type: This is an exponential function because the variable tt is in the exponent.
    • Behavior: Since the base 1.2>11.2 > 1, the function is increasing.
      Answer: Exponential, Increasing.
  3. g(x)=14x+130g(x) = -14x + 130

    • Type: This is a linear function because the variable xx is to the first power, and there is no exponent.
    • Behavior: The slope is 14-14, which is negative, so the function is decreasing.
      Answer: Linear, Decreasing.
  4. h(x)=220(0.9)xh(x) = 220(0.9)^x

    • Type: This is an exponential function because the variable xx is in the exponent.
    • Behavior: Since the base 0.9<10.9 < 1, the function is decreasing.
      Answer: Exponential, Decreasing.
  5. f(x)=28x50f(x) = 28x - 50

    • Type: This is a linear function because the variable xx is to the first power, and there is no exponent.
    • Behavior: The slope is 2828, which is positive, so the function is increasing.
      Answer: Linear, Increasing.

Do you want more details on how to analyze such functions?

Here are 5 related questions that might interest you:

  1. What makes a function linear or exponential?
  2. How do you determine if an exponential function is increasing or decreasing?
  3. Can a linear function have a curve?
  4. What real-world phenomena can be modeled by exponential functions?
  5. How do you find the slope of a linear function?

Tip: In exponential functions, the base determines growth or decay—if the base is greater than 1, the function grows (increases); if the base is between 0 and 1, it decays (decreases).

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Math Problem Analysis

Mathematical Concepts

Linear Functions
Exponential Functions
Function Behavior (Increasing/Decreasing)

Formulas

Exponential function: y = a(b)^x
Linear function: y = mx + b

Theorems

An exponential function is increasing if the base is greater than 1 and decreasing if the base is between 0 and 1.
A linear function is increasing if the slope (m) is positive and decreasing if the slope is negative.

Suitable Grade Level

Grades 9-11