Algebra 2 Benchmark #3 - Equations Flashcards

Questions and Answers

What is the purpose of the change of base formula?

To determine the slope of a line

To calculate the area of a circle

To change a logarithm with one base to a logarithm to another base (correct)

To find the maximum height of a projectile

What is the equation of a circle?

(x-h)²+(y-k)²=r²

What is the formula for compound interest?

A=P(1+r/n)^(nt)

What is the half-life formula?

y = C (1/2) ^(t/h) Signup and view all the answers

What is the doubling formula?

y = C (2)^(t/T) Signup and view all the answers

What is the formula for growing or compounding continuously?

A = Pe^(rt) Signup and view all the answers

What is the exponential decay formula?

y = C (1-r)^(t/T) Signup and view all the answers

What is the exponential growth formula?

y = C (1+r)^(t/T) Signup and view all the answers

What is the sine of an angle θ represented as?

y/r Signup and view all the answers

What is the cosine of an angle θ represented as?

x/r Signup and view all the answers

What is the tangent of an angle θ represented as?

y/x Signup and view all the answers

What is the cosecant of an angle θ represented as?

r/y Signup and view all the answers

What is the secant of an angle θ represented as?

r/x Signup and view all the answers

What is the cotangent of an angle θ represented as?

x/y Signup and view all the answers

Is sin θ/cos θ equivalent to tan θ?

True Signup and view all the answers

Is cos θ/sin θ equivalent to cot θ?

True Signup and view all the answers

What is the average rate of change formula?

change in y/change in x Signup and view all the answers

What is the period equation?

Period = 2π / frequency Signup and view all the answers

What is a reference angle?

The acute angle formed by the terminal side of an angle in standard position and the x-axis Signup and view all the answers

What is amplitude?

The height of a wave's crest Signup and view all the answers

What is the midline in relation to a periodic function?

A horizontal axis that is used as the reference line about which the graph oscillates Signup and view all the answers

Is a negative angle generated by a clockwise rotation?

True Signup and view all the answers

Is a positive angle generated by counterclockwise rotation?

True Signup and view all the answers

What is a phase shift?

A horizontal shift for a periodic function Signup and view all the answers

What is frequency?

The number of complete waves that pass a given point in a certain amount of time Signup and view all the answers

Study Notes

Logarithmic and Exponential Formulas

Change of Base Formula: Transforms logarithmic expressions from one base to another for easier calculations.
Compound Interest Formula: Used to calculate accumulated interest on an investment; expressed as ( A = P(1 + \frac{r}{n})^{nt} ), where ( A ) is the amount, ( P ) the principal, ( r ) the interest rate, ( n ) the number of times interest is compounded per year, and ( t ) the number of years.
Half-Life Formula: Determines the remaining quantity after a certain time based on its half-life; given by ( y = C (1/2)^{t/h} ), where ( C ) is the initial amount, ( t ) is time, and ( h ) is the half-life duration.
Doubling Formula: Describes exponential growth, indicating that quantity doubles over time; modeled as ( y = C (2)^{t/T} ), where ( C ) is initial amount, ( t ) is time, and ( T ) is doubling time.
Continuous Growth Formula: Represents growth compounding continuously, formulated as ( A = Pe^{rt} ) with ( P ) as the initial investment, ( r ) the growth rate, and ( t ) time.
Exponential Decay Formula: Used in contexts of decline; represented by ( y = C (1 - r)^{t/T} ) where terms follow as in previous formulas.
Exponential Growth Formula: Indicates growth based on fixed intervals, given as ( y = C (1 + r)^{t/T} ) following similar variables.

Circle and Trigonometric Concepts

Equation of a Circle: Standard form is ( (x-h)^2 + (y-k)^2 = r^2 ); defines a circle with center ( (h, k) ) and radius ( r ).
Trigonometric Functions:
- Sine: Defined as ( \sin θ = \frac{y}{r} ).
- Cosine: Defined as ( \cos θ = \frac{x}{r} ).
- Tangent: Expressed as ( \tan θ = \frac{y}{x} ).
- Cosecant: Inversely related to sine; ( \csc θ = \frac{r}{y} ).
- Secant: Inversely related to cosine; ( \sec θ = \frac{r}{x} ).
- Cotangent: Inversely related to tangent; ( \cot θ = \frac{x}{y} ).

Trigonometric Identities

Tangent Identity: ( \tan θ = \frac{\sin θ}{\cos θ} ).
Cotangent Identity: ( \cot θ = \frac{\cos θ}{\sin θ} ).

Additional Concepts

Average Rate of Change: Calculated as the ratio of the change in ( y ) to the change in ( x ).
Period Equation: Represents the duration of one complete cycle in periodic functions, given by ( \text{Period} = \frac{2π}{\text{frequency}} ).
Reference Angle: The acute angle formed between the terminal side of an angle in standard position and the x-axis.
Amplitude: Maximum height of a wave's crest from its midline.
Midline: A horizontal axis serving as a reference for the oscillation of periodic function graphs.
Negative Angle: Refers to angles measured in a clockwise direction.
Positive Angle: Refers to angles measured in a counterclockwise direction.
Phase Shift: A horizontal displacement in the graph of a periodic function.
Frequency: The count of complete waves passing a specific point in a designated time frame.

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Test your knowledge on key algebra concepts with this set of flashcards. This quiz focuses on important equations such as the change of base formula, the equation of a circle, and formulas for compound interest and half-life. Perfect for reviewing essential math principles in Algebra 2.