Algebra 2 Chapter 7 Flashcards

Questions and Answers

What is the equation for direct variation between two variables x and y?

y = x/k

y = k/x

y = kx (correct)

y = k

What is the equation for inverse variation between two variables x and y?

y = k + x

y = k/x (correct)

y = kx

y = k - x

Does the data (x: 6.5, 13, 104; y: 8, 4, 0.5) represent direct or inverse variation?

inverse variation

Does the data (x: 5, 8, 12; y: 30, 48, 72) represent direct or inverse variation?

direct variation Signup and view all the answers

Identify if the equation xy = 17 is inverse or direct variation.

Inverse variation Signup and view all the answers

Identify if the equation y = x + 29 is inverse or direct variation.

Neither Signup and view all the answers

Identify if the equation y/10 = x is inverse or direct variation.

Direct variation Signup and view all the answers

What is the value of k if y = 27 when x = 6 in direct variation?

4.5 Signup and view all the answers

What is the value of d when e = 10 if e varies directly as d and e = 3.85 when d = 50?

approximately 13 Signup and view all the answers

What is the equation for inverse variation if y = 4 when x = 5?

y = 20/x Signup and view all the answers

What is the time t needed to complete a race if t varies inversely with the runner's speed s, and t = 2.97 hours when s = 8.82 m/h?

t = 26.1954/s Signup and view all the answers

What is the relationship called between three or more variables that can be written as y = kxz?

joint variation Signup and view all the answers

If the volume of a cone varies jointly with the base B and height H, what is the equation if V = 12 pi when B = 9 pi and H = 4?

v = (1/3)bh Signup and view all the answers

What type of variation combines both direct and inverse variations?

combined variation Signup and view all the answers

What is the function represented by the equation f(x) = 1/x?

rational function Signup and view all the answers

What is the vertical asymptote for the function g(x) = 1/x + 2?

x = -2 Signup and view all the answers

Match the following terms to their definitions:

Rational Function = Function represented by the ratio of two polynomials Vertical Asymptote = x = value that makes the function undefined Horizontal Asymptote = y approaches a constant value at infinity Domain = Set of all possible x-values for a function Signup and view all the answers

What is the vertical asymptote for the function ax+b/cx+d?

x = -d/c Signup and view all the answers

What is the least common multiple of 4x^2y^3 and 6x^4y^5?

12x^4y^5 Signup and view all the answers

What is the least common denominator for x^2-2x-3 and x^2-x-6?

(x-3)(x+2)(x+1) Signup and view all the answers

What is the common denominator when adding (x-3)/(x^2+3x-4) + 2x/(x+4)?

(x+4)(x-1) Signup and view all the answers

What is the simplified version of (2x^2-30)/(x^2-9) - (x+5)/(x+3)?

(x-5)/(x-3) Signup and view all the answers

State when x is undefined for the expression (x-3)/(x^2+3x-4).

-4 Signup and view all the answers

What is an important note to remember while working on expressions?

Check over your work. Signup and view all the answers

Study Notes

Direct and Inverse Variation

Direct variation defined as a relationship where y = kx, with k ≠ 0; constant ratio indicating y varies directly with x.
Inverse variation defined as y = k/x, with k ≠ 0; constant product indicating y varies inversely with x.
Data analysis to determine variation type:
- Inverse when a constant product is observed (e.g., {6.5, 13, 104} with corresponding y values yielding a product of 52).
- Direct when a constant ratio is maintained (e.g., {5, 8, 12} with corresponding y values yielding a ratio of 6).

Solving and Graphing Variations

To solve direct variation problems, identify k by rearranging y = kx. Example: if y = 27 when x = 6, then k = 4.5, leading to y = 4.5x.
Word problems can be modeled with equations such as e = kd to find missing variables (e.g., d calculated when e = 10 yields approximately 13).
For inverse variations, determine y = k/x with known values to find k (e.g., k = 20 when y = 4 and x = 5).

Joint and Combined Variation

Joint variation occurs when y varies jointly with multiple variables (e.g., y = kxz).
Combined variation incorporates both direct and inverse relationships; generally expressed as a fraction where varying quantities are in the numerator and denominators.

Rational Functions

Rational function denoted by f(x) = 1/x; its graph forms a hyperbola with vertical and horizontal asymptotes.
Transformed rational functions take the form f(x) = (a/(x-h)) + k, indicating reflections, shifts, and vertical adjustments.
Domain excludes vertical asymptotes, and range excludes horizontal asymptotes.

Graphing and Analyzing Rational Functions

To graph rational functions, identify key points and corresponding asymptotes.
Finding the least common denominator (LCD) is crucial for adding or subtracting rational expressions involving unlike denominators.

Simplifying Rational Expressions

Simplification involves factoring numerators and denominators, cancelling common factors, and re-evaluating undefined values based on original denominators.
Processes of multiplying and dividing rational expressions include flipping the divisor and simplifying.

Adding and Subtracting Rational Expressions

When combining fractions, ensure a common denominator is established; retain factors from each expression when necessary.
Subtraction also requires a common denominator, and careful assembly of terms is important to avoid errors.

Key Study Tips

Verify each step when simplifying or manipulating rational expressions.
Pay attention to where x may be undefined; this includes values that result in zero denominators.
Practice restructuring equations and graphical interpretations to enhance understanding.

Encouragement

Stay confident; consistent practice helps build mastery in algebraic expressions and rational functions.

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Description

Test your knowledge of direct and inverse variations in Algebra 2 Chapter 7. This quiz covers key concepts including the relationships between variables and their mathematical definitions. Prepare to master the essential principles of variation with these flashcards.