Algebra 2 Unit 8 Flashcards

Questions and Answers

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.

horizontally

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 ____.

base Signup and view all the answers

The base of the logarithm function can be equal to one.

False Signup and view all the answers

How much of a 50 gram sample of Thorium-228 will exist after 22.8 years?

0.124 grams Signup and view all the answers

The number, ____, so that $b^y=x$ is called the logarithm of x.

y Signup and view all the answers

If $ ext{log }5=0.83$, then $ ext{log }7$ is equal to ____.

1.20 Signup and view all the answers

If $ ext{in }3=1.10$ and $ ext{in }6=1.79$, then $ ext{in }2$ is equal to ____.

0.69 Signup and view all the answers

After 5 years, how much would an investment of $7,000 at 3% interest, compounded semi-annually, be worth?

$8,123.79 Signup and view all the answers

For an exponential function $f(x)=ab^cx$, changing the value for ____ will change the y-intercept.

a Signup and view all the answers

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.

Description

Test your knowledge with these flashcards focused on key concepts from Algebra 2 Unit 8. Each card includes essential terms and definitions, covering topics such as exponential functions and graph transformations. Perfect for quick revision before exams!