Algebra Class: Sequences and Functions
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Questions and Answers

Given an arithmetic sequence where the first term is -8 and the common difference is 6, what is the explicit formula?

  • $a_n = 6 - 8n$
  • $a_n = -8 + 6n$
  • $a_n = -2 + 6n$
  • $a_n = -14 + 6n$ (correct)

In the arithmetic sequence 7, 4, 1, ..., what is the value of the seventh term ($a_7$)?

  • -11
  • -14
  • -17
  • -8 (correct)

A piecewise function is defined as follows: $f(x) = \begin{cases} 3x - 1, & x < 1 \ x + 2, & x \geq 1 \end{cases}$. What is the value of f(-2)?

  • -7 (correct)
  • -5
  • 4
  • 0

A dataset of Olympic 500-meter Gold Medal Speed Skating times is analyzed. Which of the following is a valid interpretation regarding causality?

<p>The data does not provide enough information to confirm a causal relationship, there might be other influencing factors. (B)</p> Signup and view all the answers

Given $p(x) = \frac{10}{x-3}$, what is the value of $p(5)$?

<p>5 (C)</p> Signup and view all the answers

If $p(x) = \frac{10}{x-3}$ and $p(x) = 13$, what is the value of x?

<p>$\frac{49}{13}$ (B)</p> Signup and view all the answers

A function $f(x)$ is shifted 2 units up, resulting in a new function $g(x)$. What change will happen to the original function $f(x)$?

<p>$g(x) = f(x) + 2$ (D)</p> Signup and view all the answers

A data set shows the number of graduating 8th graders from a middle school since 2012. Using linear regression analysis, a correlation coefficient of 0.92 is found. Which statement below accurately interprets this value?

<p>There is a strong positive linear relationship between the year and the number of graduating 8th graders. (D)</p> Signup and view all the answers

Which of the following relations is a function?

<p>{(2,7), (6,5), (4,4), (5,1), (3,3)} (C)</p> Signup and view all the answers

A function is defined as $y = 4x + 3$. Which of the following points lies on its graph?

<p>(1, 7) (B)</p> Signup and view all the answers

Ms. Yi tutors after work for x hours. Which is the most reasonable domain for the 'x' in this context?

<p>0 &lt; x &lt; 5 (B)</p> Signup and view all the answers

Gary starts with 25 Pokemon cards and gets 3 more each week. Given n represents the number of weeks, which explicit formula describes this?

<p>$a_n = 25 + 3(n-1)$ (C)</p> Signup and view all the answers

Given the mapping: {-7 -> 2, 0 -> 2, 3 -> 1}, what is the range of this relation?

<p>{1, 2} (D)</p> Signup and view all the answers

Given the mapping: {-7 -> 2, 0 -> 2, 3 -> 1}, is this relation a function?

<p>Yes, because each input has only one output. (B)</p> Signup and view all the answers

Which of the following equations represents a line with a slope of $1/4$ ?

<p>$f(x) = \frac{1}{4}x - 3$ (A)</p> Signup and view all the answers

What is the y-intercept of the line represented by the equation $f(x) = \frac{1}{4}x - 3$?

<p>-3 (A)</p> Signup and view all the answers

Which of the following relations represents a function?

<p>{ (2, 7), (6, 5), (4, 4), (5, 1), (3, 3) } (C)</p> Signup and view all the answers

Which function corresponds to the table with values (0,-5), (1,-3), (2,-1), (3,1)?

<p>$y = \frac{1}{2}x - 5$ (A)</p> Signup and view all the answers

Ms. Yi tutors her friends' children for x hours after school on Wednesdays. Which of the following is the most reasonable domain for x?

<p>$0 &lt; x &lt; 5$ (B)</p> Signup and view all the answers

Gary starts with 25 Pokémon cards and gets 3 new cards per week. Which arithmetic sequence describes the number of cards he has after n weeks?

<p>$a_n = 25 + 3n$ (A)</p> Signup and view all the answers

Given a mapping with the following ordered pairs: {(1, 2), (2, 4), (3, 6), (4, 8)}, what is the corresponding range?

<p>{2, 4, 6, 8} (B)</p> Signup and view all the answers

For the function $f(x) = \frac{1}{2}x - 3$, what is the value of $f(6)$?

<p>0 (C)</p> Signup and view all the answers

A function is defined by the rule $y = 4x - 7$. If the input x is 5, what is output y?

<p>13 (D)</p> Signup and view all the answers

Which of the following is the explicit formula for an arithmetic sequence?

<p>$a_n = a_1 + d(n-1)$ (C)</p> Signup and view all the answers

Given the function $f(x)$, where $f(-6) = -12$ and $f(4) = 3$, and $g(x)$ is $f(x)$ shifted 2 units up, what is the function for $g(x)$?

<p>$g(x) = x - 1$ (C)</p> Signup and view all the answers

What is the explicit formula, using distribution, for the arithmetic sequence -8, -2, 4, 10, ...?

<p>$A(n) = 6n - 14$ (B)</p> Signup and view all the answers

Given the piecewise function: $f(x) = \begin{cases} 3x^2 - 1, & x < 1 \ 0, & x = 1 \ x + 2, & x \geq 1 \end{cases}$, find $f(-2)$.

<p>11 (A)</p> Signup and view all the answers

Based on the provided information, which best describes the correlation of the Olympic 500-meter Gold Medal Speed Skating times?

<p>Negative correlation (D)</p> Signup and view all the answers

Given the function $p(x) = \frac{x}{8}+ \frac{3}{4}$, what is the value of $p(5)$?

<p>$4$ (D)</p> Signup and view all the answers

For the function $p(x) = \frac{x}{8} + \frac{3}{4}$, if $p(x) = 13$, what is the value of x?

<p>32 (D)</p> Signup and view all the answers

A set of data shows the number of graduating 8th graders from a middle school since 2012. The linear regression equation is $y = 9.17x - 18114.68$, and the correlation coefficient is 0.99. What does the correlation coefficient mean in this context?

<p>There is a strong positive relationship between year and number of graduates. (A)</p> Signup and view all the answers

Flashcards

f(-6)

The value of the function f(x) when x = -6.

f(4)

The value of the function f(x) when x = 4.

Domain of f(x)

The set of all possible input values (x) for which the function is defined.

Range of f(x)

The set of all possible output values (y) for the function.

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Arithmetic Sequence

A sequence where the difference between any two consecutive terms is constant.

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Explicit Formula for Arithmetic Sequence

A formula that defines the nth term of an arithmetic sequence in terms of its position (n) and the first term (a1) and the common difference (d).

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Function Evaluation

The method used to calculate the value of a function for a given input value.

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Causal Relationship

A relationship between two variables where one variable directly causes a change in the other.

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Domain of a function

The set of all possible input values (x-values) for a function.

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Range of a function

The set of all possible output values (y-values) for a function.

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Linear function

A function that can be represented by a straight line. Its equation is in the form y = mx + c, where 'm' is the slope and 'c' is the y-intercept.

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Correlation coefficient

A measure of the strength and direction of the linear relationship between two variables.

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Linear regression equation

An equation that best fits the data points in a scatter plot. It can be used to predict the value of one variable given the value of another.

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Piecewise function

A function defined by multiple expressions, each with a specific domain. The function uses the expression corresponding to the input value's domain.

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What is a function?

A relation is a function if each input (x-value) has exactly one output (y-value). This means no x-values can be repeated with different y-values.

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What is domain and range?

The domain is the set of all possible input values (x-values) for a function. The range is the set of all possible output values (y-values).

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What is an arithmetic sequence?

In an arithmetic sequence, each term is found by adding a constant value (the common difference) to the previous term. The explicit formula allows us to find any term in the sequence directly.

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What is the explicit formula for an arithmetic sequence?

The explicit formula for an arithmetic sequence is: an = a1 + (n - 1) d Where an is the nth term, a1 is the first term, n is the term number, and d is the common difference.

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How do we determine the domain in a real-world scenario?

The domain represents the possible values that the independent variable (often 'x') can take. In this case, the domain is limited by the context: Ms. Yi can't tutor for negative hours or more than 24 hours in a day.

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How can we tell if a mapping is a function?

A function is a special type of relation where each input has exactly one output. A mapping represents a relation, and we can determine if it's a function by checking if any input has more than one output.

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How can we represent a real-world scenario with an arithmetic sequence?

An arithmetic sequence can be used to represent situations involving a constant change, like Gary's Pokémon card collection. The starting number (a1) is 25, and the constant increase (d) is 3 cards per week, so the sequence is 25, 28, 31, 34... (a1 + d, a1 + 2d, a1 + 3d...) .

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How can we determine ordered pairs and if a mapping is a function?

To list the ordered pairs from a mapping, take each input (x-value) and pair it with its corresponding output (y-value). Check for any repeating x-values with different y-values, which would indicate a relation that is not a function.

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What are domain and range?

The domain of a function is the set of all possible input values (x-values). The range is the set of all possible output values (y-values).

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Is {(2, 7), (6, 5), (4, 4), (5, 1), (3, 3)} a function?

To determine if a relation is a function, check if each input (x-value) has only one output (y-value). In the given relation, (2, 7), (6, 5), (4, 4), (5, 1), (3, 3), each input has a unique output, making it a function.

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What is the most reasonable domain for Ms. Yi's tutoring hours? (0 < x < 48, 0 < x < 24, 0 < x < 12, 0 < x < 5)

The domain represents the possible values for the independent variable (x). In this case, the domain is restricted for hours Ms. Yi tutors after school, making 0 < x < 12 the most reasonable choice.

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Write an arithmetic sequence for Gary's Pokémon cards

An arithmetic sequence is a pattern where each term is found by adding a constant difference to the previous term. In this case, Gary starts with 25 cards (a1 = 25) and gains 3 cards each week (d = 3), so the sequence is A(n) = 25 + (n - 1)(3)

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What are the domain, range, ordered pairs and is the mapping a function - {-7, 0, 3} -> {1, 2} : (-7, 2), (0, 2), (3, 1)

The domain represents the input values (x) and the range represents the output values (y). In this mapping, the domain is {-7, 0, 3} and the range is {1, 2}. The ordered pairs are {(-7, 2), (0, 2), (3, 1)}. Each input (x) has only one output (y), so the relation is a function.

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Study Notes

Arithmetic Sequences

  • Explicit formula: an = a1 + (n − 1)d
    • an represents the nth term
    • a1 is the first term
    • n is the term number
    • d is the common difference

Functions

  • A relation is a function if each input (x-value) has only one output (y-value).
  • Verifying if a relation is a function:
    • Check if any x-values are repeated in the relation. If an x-value is repeated, and the y-values are different, the relation is not a function.

Function Tables

  • Given a function rule, a function table lists corresponding input (x) and output (y) values.

Domains

  • The domain of a function is the set of all possible input values (x-values).
  • Reasonable domains often consider practical limitations or restrictions.

Arithmetic Sequences (Example)

  • If Gary starts with 25 cards and gets 3 more each week, the number of cards after n weeks can be described by an arithmetic sequence:
    • A(n) = 25 + (n - 1) * 3

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Description

This quiz covers key concepts in algebra related to arithmetic sequences and functions. You'll explore explicit formulas, function verification, function tables, and the definition of domains. Test your understanding of these essential topics in algebra!

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