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Questions and Answers
What is the product of the factors 2 and 7?
What is the product of the factors 2 and 7?
What is the sum of the numbers -9 and 2?
What is the sum of the numbers -9 and 2?
What is the general form of a rational expression?
What is the general form of a rational expression?
Which step is crucial when multiplying rational expressions?
Which step is crucial when multiplying rational expressions?
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What do you need to find when adding or subtracting rational expressions?
What do you need to find when adding or subtracting rational expressions?
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What does the radical sign (√) denote?
What does the radical sign (√) denote?
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What must be checked after performing addition or subtraction on rational expressions?
What must be checked after performing addition or subtraction on rational expressions?
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What is the product of conjugates?
What is the product of conjugates?
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What is the prime factorization of 180?
What is the prime factorization of 180?
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Which exponent rule allows you to simplify $a^m \cdot a^n$?
Which exponent rule allows you to simplify $a^m \cdot a^n$?
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If $x^0 = 1$, then what would $5^0$ equal?
If $x^0 = 1$, then what would $5^0$ equal?
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How can a relation be described?
How can a relation be described?
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What does the domain of a relation represent?
What does the domain of a relation represent?
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When simplifying a power of a product, what is the resulting expression for $(xy)^n$?
When simplifying a power of a product, what is the resulting expression for $(xy)^n$?
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Which of the following statements about negative exponents is correct?
Which of the following statements about negative exponents is correct?
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What happens to the graph when the vertical stretch factor, a, is less than 0?
What happens to the graph when the vertical stretch factor, a, is less than 0?
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Which value indicates the vertical translation of a sine or cosine function?
Which value indicates the vertical translation of a sine or cosine function?
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If k < 0, how is the graph affected?
If k < 0, how is the graph affected?
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In the equation $g(x) = 0.5 \cdot \sin(x + 45)$, what is the effect of the coefficient 0.5?
In the equation $g(x) = 0.5 \cdot \sin(x + 45)$, what is the effect of the coefficient 0.5?
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What does a positive value of p indicate regarding horizontal translation?
What does a positive value of p indicate regarding horizontal translation?
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What does the parameter 'b' in trigonometric functions represent?
What does the parameter 'b' in trigonometric functions represent?
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When applying transformations to a graph, which transformation is performed last?
When applying transformations to a graph, which transformation is performed last?
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For the function $f(x) = cos(x)$, what is the maximum value it can achieve?
For the function $f(x) = cos(x)$, what is the maximum value it can achieve?
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If q < 0, what effect does this have on the graph?
If q < 0, what effect does this have on the graph?
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In what scenario does a parabola open downward based on its leading coefficient?
In what scenario does a parabola open downward based on its leading coefficient?
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For the graph of the function f(x), if the vertical stretch factor a is equal to 2, what is true?
For the graph of the function f(x), if the vertical stretch factor a is equal to 2, what is true?
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Which transformation occurs first when transforming a graph?
Which transformation occurs first when transforming a graph?
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If the graph is reflected in both the x-axis and y-axis, what does this imply about the values of a and k?
If the graph is reflected in both the x-axis and y-axis, what does this imply about the values of a and k?
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What does the domain of an inverse function correspond to in the original function?
What does the domain of an inverse function correspond to in the original function?
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Which statement is true about the graph of an inverse function?
Which statement is true about the graph of an inverse function?
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What must be done to ensure that the inverse of a function is also a function?
What must be done to ensure that the inverse of a function is also a function?
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If $y = f(x)$, what is the first step in determining the inverse function?
If $y = f(x)$, what is the first step in determining the inverse function?
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Under which condition will the inverse of a function not be a function?
Under which condition will the inverse of a function not be a function?
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What effect does the coefficient 'a' have in transformations of functions?
What effect does the coefficient 'a' have in transformations of functions?
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What must be evaluated when you find the value of a function's inverse at a specific point?
What must be evaluated when you find the value of a function's inverse at a specific point?
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When sketching the graph of an inverse function, which of the following steps is necessary?
When sketching the graph of an inverse function, which of the following steps is necessary?
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Study Notes
Factoring Polynomials
- Product can be expressed as two factors, e.g., (14 = 2 \times 7) and (–18 = 3 \times (–6)).
- Sum used in factoring is defined as the result of added factors, e.g., (9 = 2 + 7) and (-7 = -9 + 2).
- Decompose the middle term for factoring, e.g., (–7xy) can be written as (–9xy + 2xy).
- Factor by grouping as a common method.
- Identify special factoring patterns, such as perfect square trinomials and difference of squares.
Rational Expressions
- Rational expressions are formed as a ratio of two polynomials, (F) and (G).
- Simplification involves factoring both the numerator and denominator, while noting restrictions to the variables.
- In multiplication and division, invert and multiply, then simplify while considering new restrictions.
- For addition and subtraction, find the least common denominator (LCD), combine the rational expressions, then simplify where possible.
Radicals
- A radical sign denotes the root of a number, with (n) as the index and (a) as the radicand.
- When conjugates are multiplied, the result yields a rational expression with no radicals.
- Prime factorization simplifies numbers into their basic prime components using methods like tree diagrams.
Exponent Rules
- Product Rule: (a^m \times a^n = a^{m+n})
- Quotient Rule: (a^m / a^n = a^{m-n})
- Power of a Power: ((a^m)^n = a^{mn})
- Power of a Quotient: ((a/b)^n = a^n/b^n)
- Zero Exponents: (a^0 = 1) for (a \neq 0)
- Negative Exponents: (a^{-n} = 1/a^n)
- Rational Exponents provide roots in exponent form, e.g., (a^{1/n}) is the (n)-th root of (a).
Functions
- A relation connects two sets and can be expressed through equations, diagrams, graphs, or tables.
- Domain refers to all possible input values (x-values) in a function.
- Inverse Functions map outputs back to their corresponding inputs, with the domain of the inverse being the range of the original function.
- To evaluate an inverse function, interchange (x) and (y) and solve for (y).
Graph Transformations
- Parameters (a), (k), (p), and (q) impact the graph's shape and position:
- (a) stretches vertically; (k) stretches horizontally.
- (p) shifts the graph left or right; (q) shifts up or down.
- Stretches and reflections are applied before translations.
Quadratic Functions
- The graph of (y = ax^2 + bx + c) is a parabola:
- Opens upward if (a > 0) and downward if (a < 0).
- The value of (a) determines the parabola's maximum or minimum point.
- The period for trigonometric functions is affected by the coefficient (k) in (y = a \sin(kx)) or (y = a \cos(kx)).
Overall Exam Review Tips
- Understand each concept's definitions and properties.
- Practice factoring, simplifying rational expressions, and graphing transformations.
- Familiarize yourself with evaluating functions and their inverses.
- Solve problems involving exponent rules and their applications.
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Description
In this quiz, you will explore the techniques for factoring polynomials through various examples and methods. It specifically focuses on factoring by grouping and understanding the relationships between product and sum. Test your skills and see how well you can factor different polynomial expressions!
