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Questions and Answers
What is the common ratio of the sequence -2, 6, -18, 54?
What is the common ratio of the sequence -2, 6, -18, 54?
-3
What value should Yui use as the common ratio for her savings account balances which increase by 5% each month?
What value should Yui use as the common ratio for her savings account balances which increase by 5% each month?
1.05
What is the initial value of the exponential function shown on the graph?
What is the initial value of the exponential function shown on the graph?
4
Which function represents g(x), a reflection of f(x) = 1/2(3)x across the y-axis?
Which function represents g(x), a reflection of f(x) = 1/2(3)x across the y-axis?
Which table of values could be used to graph g(x), a reflection of f(x) = One-third(6)x across the x-axis?
Which table of values could be used to graph g(x), a reflection of f(x) = One-third(6)x across the x-axis?
What is the initial value of the exponential function shown on the graph?
What is the initial value of the exponential function shown on the graph?
Which best describes the range of the function f(x) = 2/3(6)x after it has been reflected over the x-axis?
Which best describes the range of the function f(x) = 2/3(6)x after it has been reflected over the x-axis?
Which is a stretch of an exponential decay function?
Which is a stretch of an exponential decay function?
Which function represents a reflection of f(x) = 5(0.8)x across the x-axis?
Which function represents a reflection of f(x) = 5(0.8)x across the x-axis?
Which formula can be used to describe the sequence 1.2, 3, 7.5, 18.75,...?
Which formula can be used to describe the sequence 1.2, 3, 7.5, 18.75,...?
Which graph is the sequence defined by the function f(x) = 3(2)x-1?
Which graph is the sequence defined by the function f(x) = 3(2)x-1?
What is the initial value of the exponential function represented by the table?
What is the initial value of the exponential function represented by the table?
Which exponential function is represented by the graph?
Which exponential function is represented by the graph?
Which graph represents a function with a growth factor of 5?
Which graph represents a function with a growth factor of 5?
Which function represents g(x), a reflection of f(x) = 6(1/3)x across the y-axis?
Which function represents g(x), a reflection of f(x) = 6(1/3)x across the y-axis?
Which function is a shrink of the exponential growth function shown on the graph?
Which function is a shrink of the exponential growth function shown on the graph?
Which is the graph of the sequence defined by the function f(x + 1) = 2/3f(x) if the initial value of the sequence is 108?
Which is the graph of the sequence defined by the function f(x + 1) = 2/3f(x) if the initial value of the sequence is 108?
Which graph represents a function with an initial value of 1/2?
Which graph represents a function with an initial value of 1/2?
Which exponential function is represented by the table?
Which exponential function is represented by the table?
When writing an explicit formula to represent beach erosion at a rate of 1.1% each year, which value should be used as the common ratio?
When writing an explicit formula to represent beach erosion at a rate of 1.1% each year, which value should be used as the common ratio?
What explicit formula can Sebastian use to model his geometric sequence whose second term is 12?
What explicit formula can Sebastian use to model his geometric sequence whose second term is 12?
Which ordered pair is on g(x) when the function f(x) = 1/6 (2/5)x is reflected across the y-axis?
Which ordered pair is on g(x) when the function f(x) = 1/6 (2/5)x is reflected across the y-axis?
Which is the range of the function f(x) = 1/7(9)x?
Which is the range of the function f(x) = 1/7(9)x?
Which function represents g(x), a reflection of f(x) = 4(1/2)x across the x-axis?
Which function represents g(x), a reflection of f(x) = 4(1/2)x across the x-axis?
Consider the exponential function f(x) = 3(1/3)x and its graph. Which statements are true for this function and graph? Select three options.
Consider the exponential function f(x) = 3(1/3)x and its graph. Which statements are true for this function and graph? Select three options.
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Study Notes
Common Ratios and Sequences
- The common ratio of the sequence -2, 6, -18, 54... is -3.
- For a recursive formula for savings account balances increasing by 5% monthly, the common ratio is 1.05.
Exponential Functions
- The initial value of the exponential function can vary: one example shows an initial value of 4, while another shows an initial value of 2.
- A reflection across the y-axis for the function f(x) = (1/2)(3^x) results in g(x) = (1/2)(3^(-x)).
- The function f(x) = (1/3)(6^x) reflects across the x-axis as g(x) by taking the negatives of the outputs.
Graphs and Their Properties
- The reflected function f(x) = (2/3)(6^x) across the x-axis has a range of all real numbers less than or equal to 0.
- A stretch of an exponential decay function is represented by f(x) = (5/4)(4/5)^x, indicating a growth in the decay rate.
- g(x) = -5(0.8^x) represents a reflection of the exponential growth function f(x) = 5(0.8^x) across the x-axis.
Formulas and Representation
- A sequence defined as 1.2, 3, 7.5, 18.75... can be described by the formula f(x) = 1.2(2.5)^(x-1).
- The initial value of the sequence represented by the function f(x) = (3)(2^x-1) has specific graph coordinates that can be identified (e.g., (1, 3), (2, 6)).
- A common ratio for a beach eroding at a rate of 1.1% per year is calculated as 0.989.
Recursive and Explicit Formulas
- The recursive formula f(x+1) = 4f(x) with a second term of 12 leads to the explicit formula f(x) = 3(4)^(x-1).
- Reflecting the function f(x) = (1/6)(2/5)^x across the y-axis yields ordered pairs such as (-3, 4/375).
General Characteristics of Functions
- The range of the function f(x) = (1/7)(9^x) encompasses all real numbers greater than 0.
- The function f(x) = 3(1/3)^x depicts exponential decay and represents a stretch of the function f(x) = (1/3)^x, illustrating decreasing values over time.
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