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Questions and Answers
What does it mean to shift the graph of $f(x)=a+c ext{log}(dx+g)$ vertically?
What does it mean to shift the graph of $f(x)=a+c ext{log}(dx+g)$ vertically?
- Change parameter d
- Change parameter a (correct)
- Change parameter c (correct)
- Change parameter g
To shift the graph of $f(x)=a+c ext{log}(b(dx+g))$ _____, change the parameter g.
To shift the graph of $f(x)=a+c ext{log}(b(dx+g))$ _____, change the parameter g.
horizontally
What is a function of the form $f(x)=b^x$ always goes through?
What is a function of the form $f(x)=b^x$ always goes through?
(1, b) and (0, 1)
For an exponential function $f(x)=ab^cx$, changing the value for c will change the ____.
For an exponential function $f(x)=ab^cx$, changing the value for c will change the ____.
The base of the logarithm function can be equal to one.
The base of the logarithm function can be equal to one.
How much of a 50 gram sample of Thorium-228 will exist after 22.8 years?
How much of a 50 gram sample of Thorium-228 will exist after 22.8 years?
The number, ____, so that $b^y=x$ is called the logarithm of x.
The number, ____, so that $b^y=x$ is called the logarithm of x.
If $ ext{log }5=0.83$, then $ ext{log }7$ is equal to ____.
If $ ext{log }5=0.83$, then $ ext{log }7$ is equal to ____.
If $ ext{in }3=1.10$ and $ ext{in }6=1.79$, then $ ext{in }2$ is equal to ____.
If $ ext{in }3=1.10$ and $ ext{in }6=1.79$, then $ ext{in }2$ is equal to ____.
After 5 years, how much would an investment of $7,000 at 3% interest, compounded semi-annually, be worth?
After 5 years, how much would an investment of $7,000 at 3% interest, compounded semi-annually, be worth?
For an exponential function $f(x)=ab^cx$, changing the value for ____ will change the y-intercept.
For an exponential function $f(x)=ab^cx$, changing the value for ____ will change the y-intercept.
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Study Notes
Graph Shifts
- Vertical graph shifts for the function ( f(x) = a + c \log(dx + g) ) result from changing parameters ( a ) or ( d ).
- Horizontal graph shifts occur by altering the parameter ( g ).
Key Exponential Concepts
- Exponential functions of the form ( f(x) = b^x ) pass through the points (1, b) and (0, 1) regardless of the base ( b ).
- Adjusting the parameter ( c ) in ( f(x) = ab^cx ) modifies the rate of growth or decay.
Logarithmic Properties
- Logarithmic functions cannot have a base equal to one and must remain positive to uphold their properties.
- The logarithm of ( x ) is defined as the number ( y ) such that ( b^y = x ).
Radioactive Decay Example
- Thorium-228 decays by 50% every 1.9 years. After 22.8 years from a 50-gram sample, approximately 0.124 grams remain.
- The decay is calculated using the formula ( A = A_0 e^{-kt} ), where ( A_0 ) is the initial amount.
Financial Mathematics
- An initial investment of $7,000 at an interest rate of 3%, compounded semi-annually for 5 years, results in an approximate value of $8,123.79 using the formula ( A = P(1 + \frac{r}{n})^{nt} ).
Logarithmic Relationships
- If ( \log 5 ) equals 0.83, then ( \log 7 ) is approximately 1.20.
- If ( \ln 3 ) equals 1.10 and ( \ln 6 ) equals 1.79, then ( \ln 2 ) is about 0.69.
Graphing Variables
- For exponential functions, changing parameter ( a ) affects the y-intercept of the graph.
- In graphing, letters ( A, b ) and ( B, a ) can denote different axes or coordinate points.
Function Analysis
- The expression ( 4 + \log x ) signifies a vertical shift upward by 4 units in relation to the logarithmic function.
- The logarithmic expression ( \log(x - 7) ) indicates a horizontal shift to the right by 7 units.
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