Piecewise part 2 worked
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This document contains homework instructions for writing piecewise functions from graphs. It explains that students should find the vertex of each graph, write the first function the same and the second function as the negative, and state the restrictions based on the vertex. Several example graphs are given for students to write as piecewise functions and graph. The document concludes by assigning a homework worksheet for practice.
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Rises to the left, falls to the right
Rises to the right, rises to the left
Falls to the left, rises to the right
Falls to the right
Falls to the left, falls to the right
4 points
QUESTION 2
1. Select the correct description of right-hand and left-hand behavior of the graph of the polynomial funciton.
f(x) = 4x2 - 5x + 4
Falls to the left, rises to the right
Falls to the left, falls to the right
Rises to the left, rises to the right
Rises to the left, falls to the right
Falls to the left
4 points
QUESTION 3
1. Find all the real zeroes of the polynomial function.
f(x) = x2 - 25
-25
5
-5
25
±5
4 points
QUESTION 4
1. Use synthetic division to divide.
(4x3 + x2 - 11x + 6) ÷ (x + 2)
4x2 - 5x - 6
4x2 - 7x + 3
4x2 - 2x - 2
4x2 + 5x - 12
4x2 + 7x - 4
4 points
QUESTION 5
1. Use the Remainder Theorem and synthetic division to find the function value. Verify your answers using another method.
h(x) = x3 - 6x2 - 5x + 7
h(-8)
-849
-847
-851
-848
-845
4 points
QUESTION 6
1. Find all the rational zeroes of the function.
x3 - 12x2 + 41x - 42
-2, -3, -7
2, 3, 7
2, -3, 7
-2, 3, 7
-2, 3, -7
4 points
QUESTION 7
1. The total revenue R earned (in thousands of dollars) from manufacturing handheld video games is given by
R(p) = -25p2 + 1700p
where p is the price per unit (in dollars).
Find the unit price that will yield a maximum revenue.
$38
$35
$36
$37
$34
4 points
QUESTION 8
1. Find the domain of the function
Domain: all real numbers x except x = 7
Domain: all real numbers x except x = ±49
Domain: all real numbers x except x = ±8
Domain: all real numbers x except x = -7
Domain: all real numbers x except x = ±7
4 points
QUESTION 9
1. Find the domain of the function and identify any vertical and horizontal asymptotes.
Domain: all real numbers x
Vertical asymptote: x = 0
Horizontal asymptote: y = 0
Domain: all real numbers x except x = 2
Vertical asymptote: x = 0
Horizontal asymptote: y = 0
Domain: all real numbers x except x = 5
Vertical asymptote: x = 0
Horizontal asymptote: y = 2
Domain: all real numbers x
Vertical asymptote: x = 0
Horizontal asymptote: y = 2
Domain: all real numbers x except x = 5
Vertical asymptote: x = 5
Horizontal asymptote: y = 0
4 points
QUESTION 10
1. Simplify f and find any vertical asymptotes of f.
x+3; vertical asymptote: x = -3
x; vertical asymptote: none
x; vertical asymptote: x = -3
x-3; vertical asymptote: none
x2; vertical asymptote: none
4 points
QUESTION 11
1. Determine the equations of any horizontal and vertical asymptotes of
horizontal: y = 5; vertical: x = 0
horizontal: y = 1; vertical: x = -5
horizontal: y = 1; vertical: x = 1 and x = -5
horizontal: y = -1; vertical: x = -5
horizontal: y = 0; vertical: none
4 points
QUESTION 12
1. Identify all intercepts of ...
WEEK-4-Piecewise-Function-and-Rational-Function.pptx
This document discusses various topics related to piecewise functions and rational functions:
- It defines piecewise functions and provides examples of evaluating piecewise functions at given values.
- It introduces rational functions as functions of the form p(x)/q(x) where p(x) and q(x) are polynomials and q(x) is not equal to zero. It discusses representing rational functions in different forms.
- It explains how to identify restrictions or extraneous roots of rational functions by setting the denominator equal to zero. It also discusses how to determine the domain of a rational function based on its restrictions.
- Finally, it defines vertical and horizontal asymptotes of rational functions. It provides
Advance algebra
This document provides information about several advanced algebra topics:
1. The Remainder Theorem states that when dividing a polynomial f(x) by x-c, the remainder is equal to f(c).
2. The Factor Theorem states that if f(c) = 0, then x-c is a factor of the polynomial.
3. Synthetic division is a method for dividing polynomials without subtracting, by multiplying the coefficients.
4. The Rational Zeros Theorem can be used to find all possible rational zeros of a polynomial by considering factors of the leading coefficient and constant term.
Rational-Function-W3-4.pptx
The document discusses rational equations, rational inequalities, rational functions, and their intercepts, zeros, and asymptotes. It provides examples and explanations of:
- Rational equations as equations involving rational expressions
- Rational inequalities as inequalities involving rational expressions
- Rational functions as functions of the form f(x) = p(x)/q(x) where p(x) and q(x) are polynomial functions
- How to determine the x-intercepts, y-intercepts, and zeros of rational functions
- Asymptotes of rational functions as imaginary lines the graph approaches as x or y increases without limit
Relations and functions
A relation is a set of ordered pairs where the domain is the set of all first elements and the range is the set of all second elements. A function is a special type of relation where each element of the domain is mapped to exactly one element of the range. When finding the domain of a function, valid inputs are any values that do not result in division by zero or taking the square root of a negative number.
Notes - Graphs of Polynomials
The document discusses graphing and interpreting various polynomial functions using a graphing calculator. It examines the graphs of y=x-1, y=x^2-4, and y=x^3+x^2-12 to note their similarities and differences in terms of degree, number of turns, and x-intercepts. It then explores how changing the coefficients of a cubic function impacts its graph. Finally, it interprets key features of the graph of y=x^3-5x^2+x-5 such as its degree, number of turns, x-intercept, maximum and minimum points.
5.3 Graphs of Polynomial Functions
* Recognize characteristics of graphs of polynomial functions.
* Use factoring to find zeros of polynomial functions.
* Identify zeros and their multiplicities.
* Determine end behavior.
* Understand the relationship between degree and turning points.
* Graph polynomial functions.
* Use the Intermediate Value Theorem.
Relation and function pdf
The document discusses Cartesian products of sets, relations, and functions. It defines the Cartesian product as the set of all unique ordered pairs using one element from each of two sets. The number of ordered pairs in a Cartesian product equals the cardinality of the first set multiplied by the cardinality of the second set. It provides examples of calculating Cartesian products and represents them visually using grids. The document also defines relations as sets of ordered pairs and functions as special types of relations where each input is mapped to only one output. It demonstrates how to evaluate functions by substituting values for variables and find the domains of functions.
Graphing lines part 2 y = mx + b
This document discusses graphing lines using the slope-intercept form equation y=mx+b. It explains that m represents the slope and b represents the y-intercept. Examples are provided to demonstrate how to graph lines by identifying m and b values and plotting points. Students are guided in graphing several lines and then assigned independent practice graphing additional lines using the slope-intercept form.
2.4 notes b
1. Students were asked to put math problems on the board from previous homework. The document then provides examples of expressions and teaches how to simplify them using order of operations and properties like the distributive property. Students are asked to simplify sample expressions involving variables.
2. The document reviews that expressions need to have "like terms" to be simplified, such as terms with the same variables. Students practice simplifying expressions with multiple variables and terms by combining like terms.
3. To conclude, students are instructed to write their name and ID number on raffle tickets and provide just the simplified answer, practicing the skills of defining a variable, writing an expression, simplifying it, and evaluating it.
Graphing rational functions
The document discusses graphs of rational functions and how to analyze them. It covers identifying vertical and horizontal asymptotes analytically by finding values where the denominator is zero or analyzing behavior as x approaches positive/negative infinity. Examples show graphing points around asymptotes and confirming trends to sketch the graph. Key steps are finding intercepts and asymptotes, making a T-chart of values, and graphing with curves between points and asymptotes. Practice problems are provided to apply these skills.
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The document discusses simulations of chance processes using random numbers. It provides examples of simulating rolling a die, flipping a coin, using a random number table or calculator function to generate random integers, drawing cards from a deck, and generating random boy/girl outcomes from multiple births. Students are asked to simulate processes like the probability of having a boy within 5 children or collecting all cards in a trading card game to specified numbers of boxes. The document emphasizes clearly stating the question of interest, planning the simulation using an appropriate chance device, performing many repetitions, and using results to answer the original question.
Day 1
This document discusses probability and events. It defines probability as a number between 0 and 1 that describes the likelihood of an event occurring. An event is a subset of possible outcomes from a chance process. The complement of an event is the opposite outcome where the event does not happen. It provides an example of predicting which cup contains bottled water. The frequency of an event is the number of times it occurs, and relative frequency is the frequency divided by the total number of trials. It instructs the reader to flip a coin and penny 10 times each to record results and compare predictions to actual outcomes. The probability of an event can be thought of as its long-run relative frequency over many trials.
Association example
A study of 240 randomly selected college students examined the relationship between head injuries and activities. The results showed that soccer players had the highest rates of concussions, with 45 students reporting 1 concussion and 10 reporting 3 or more concussions. Using a chi-squared test of independence, the results showed a statistically significant association between number of concussions and student activities, with a p-value of 0.0015. This provided evidence to reject the null hypothesis of no association.
Homogeneity example
A study used random samples to survey the political affiliations of cell-only and landline users under 30. The study found 49% of cell-only users identified as Democrat compared to 47% of landline users. A chi-squared test was conducted to determine if phone type and political affiliation were related. The test found a p-value of 0.199897, above the standard significance level. Therefore, there is insufficient evidence to conclude that phone type and political affiliation are related for this age group.
Module 12 lesson 1 Example
The document describes how an accountant wants to determine if a client has falsified tax information by looking for patterns in invoices. The accountant inspects a random sample of 250 invoices and compares the frequencies of first digits to the probabilities predicted by Benford's Law using a chi-squared test. The results show a p-value between 0.01 and 0.005, below the significance level of 0.05. Therefore, the accountant can conclude the invoices are inconsistent with Benford's Law and suspect fraudulent activity.
Review
This document provides a study guide for a unit review that covers topics from the current and previous units, including looking over old material, identifying odd/even/neither functions, symmetry, end behavior, and determining domain and range both graphically and algebraically. Students are reminded to have their study guide ready and any questions can be asked.
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This document provides information about geometric sequences including their key properties and formulas. It defines a geometric sequence as one where the ratio between consecutive terms is constant, known as the common ratio. Formulas given include the recursive formula an = an-1r and explicit formula an = a1rn-1. Examples are provided for identifying geometric sequences and using the formulas to find missing terms. Additional notes state that arithmetic sequences have linear graphs while geometric sequences have exponential graphs.
9.4 part 1.ppt worked
This document provides information and examples about number patterns, sequence notation, recursive and explicit definitions of sequences, arithmetic sequences, and formulas for finding terms in arithmetic sequences. It includes examples of identifying arithmetic sequences, finding common differences, using recursive and explicit formulas to find specific terms, and using the arithmetic mean to find missing terms. Students are assigned a worksheet to complete problems related to these concepts but are asked to wait on questions 16-20 until they can be discussed.
Review day
The document provides information for students about an upcoming pre-calculus test, including a review of topics to study such as solving inequalities and working with different function types. It informs students that they need to create a 2-page study guide with examples and notes from each unit, which is due along with all stapled homework in order for the test.
Inverses day 2 worked
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9.4 part 4.ppt worked
1) A geometric series is the expression of the sum of terms in a geometric sequence. The sum of a finite geometric series can be calculated using the formula Sn = a(1 - r^n)/(1 - r) where a is the first term, r is the common ratio, and n is the number of terms.
2) Geometric series converge when the absolute value of the common ratio r is less than 1, and diverge when the absolute value of r is greater than 1.
3) The homework assignment includes evaluating example geometric series, determining whether other series converge or diverge, and completing a worksheet to review for a test on geometric series.
9.4 part 3.ppt worked
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Sum (S) = (First Term (a1) + Last Term (an)) * Number of Terms (n) / 2
S = a1 + (a1 + (n-1)d)
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Inverses day 1 worked
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Comp decomp worked
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Inequalities day 2 worked
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Inequalities day 1 worked
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Day 4 and 5 applications worked
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Day 3 vectors worked
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Piecewise part 2 worked
- 2. Piecewise From Graph to Equation Mrs. Ramsey
- 12. Start with the original Think about this We could rewrite it as this xy = <− ≥ = 0, 0, xx xx y
- 13. Steps 1. Find the vertex 2. First function is the same 3. Second function is the negative 4. State restrictions based on vertex
- 14. Write as a piecewise function and graph! xy 35−=
- 15. Write as a piecewise function and graph! 1y x= +
- 16. Write as a piecewise function and graph! 2 4y x= −
- 17. Write as a piecewise function and graph! 6y x= +
- 18. Write as a piecewise function and graph! 1 3y x= + +
- 19. Write as a piecewise function and graph! 2 4 5y x= − −
- 20. Write as a piecewise function and graph! 3 2y x= +