Alg2 Lesson 9 3
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This document discusses key concepts for graphing rational functions: 1) Vertical and horizontal asymptotes are determined by factors in the denominator that do or do not cancel out. 2) Discontinuities occur where the denominator equals zero, indicating holes. 3) The example function has a vertical asymptote at x= -3, a hole at x=-2, and a horizontal asymptote of y=1.
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Alg2 lesson 9-3
This rational function has:
1) A vertical asymptote at x = -3.
2) A hole at x = -2.
3) A horizontal asymptote of y = 1.
Alg2 lesson 9-3
This document discusses key concepts for graphing rational functions:
1) Vertical and horizontal asymptotes are determined by factors in the denominator that do or do not cancel out. A vertical asymptote is indicated at x = -3 and there is a hole at x = -2.
2) The horizontal asymptote is the line y = 1.
3) Examples are provided to demonstrate graphing rational functions based on identifying discontinuities and asymptotes.
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Okay, let's solve this step-by-step:
1) FOIL the expression: y = 3(x + 2)(x + 2) - 4
= 3(x^2 + 4x + 2x + 4) - 4
= 3x^2 + 12x + 12 - 4
= 3x^2 + 12x + 8
2) This is in the vertex form: y = a(x - h)^2 + k
Where: a = 3
h = 0
k = 8
So the equation in vertex form is:
y = 3(x - 0)^2 + 8
Or simply:
y = 3x^2 + 8
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Recommended
Alg2 lesson 9-3
This rational function has:
1) A vertical asymptote at x = -3.
2) A hole at x = -2.
3) A horizontal asymptote of y = 1.
Alg2 lesson 9-3
This document discusses key concepts for graphing rational functions:
1) Vertical and horizontal asymptotes are determined by factors in the denominator that do or do not cancel out. A vertical asymptote is indicated at x = -3 and there is a hole at x = -2.
2) The horizontal asymptote is the line y = 1.
3) Examples are provided to demonstrate graphing rational functions based on identifying discontinuities and asymptotes.
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The document discusses linear equations and slope. It covers plotting points on a coordinate plane, calculating slope using the rise over run formula, writing equations in slope-intercept form, finding the x- and y-intercepts, and graphing lines by making a table or using the slope and y-intercept. Methods are provided for determining the equation of a line given two points, the slope and one point, or from a graph.
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The document discusses three forms of quadratic equations - standard form, vertex form, and intercept form. It provides the key characteristics of each form and steps for graphing equations written in each form. Standard form is easiest to solve by factoring. Vertex form shows the vertex and axis of symmetry. Intercept form directly lists the x-intercepts. Converting between forms involves multiplying out terms.
5.2 solve quadratic equations by graphing.vertex and intercept form
Okay, let's solve this step-by-step:
1) FOIL the expression: y = 3(x + 2)(x + 2) - 4
= 3(x^2 + 4x + 2x + 4) - 4
= 3x^2 + 12x + 12 - 4
= 3x^2 + 12x + 8
2) This is in the vertex form: y = a(x - h)^2 + k
Where: a = 3
h = 0
k = 8
So the equation in vertex form is:
y = 3(x - 0)^2 + 8
Or simply:
y = 3x^2 + 8
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The document discusses linear equations and slope. It covers plotting points on a coordinate plane, calculating slope using the rise over run formula, writing equations in slope-intercept form, and graphing lines by making a table or using the slope and y-intercept. Methods are provided for determining the equation of a line given two points, the slope and a point, or by finding values from a graph.
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Methods3 types of functions1
Quadratic functions are polynomials of degree 2 with the general form f(x)=ax2+bx+c. The discriminant, ∆ =b2-4ac, determines the number of x-intercepts. Cubic functions are polynomials of degree 3 with the general form f(x)= ax3+bx2+cx+d. Quartic functions are polynomials of degree 4 with the general form f(x)=ax4+bx3+cx2+dx+e. Higher degree polynomial functions have more complex behaviors determined by their factors and coefficients.
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If your course has enabled Microsoft Office 365, you can upload a file from your Microsoft OneDrive for an assignment.
Like other file upload submissions, files uploaded from Office 365 are uploaded into your Canvas user files submissions folder.
Canvas accepts Microsoft Word, Microsoft PowerPoint, Microsoft Excel, and PDF types.
Notes:
If the Office 365 tab is not available in your submission, your institution has not enabled this feature.
Canvas will require you to authorize access to your OneDrive account.
In Office 365 assignments, you can only submit one file for your submission.
When Office 365 files are uploaded as a submission, later changes made to the file in OneDrive will not be updated in the submission.
If enabled in your account, Canvas plays a celebration animation when you submit an assignment on time. However, if you prefer, you can disable this feature setting in your user settings.
If the assignment you are accessing displays differently, your assignment may be using the Assignment Enhancements feature. Please view this guide for more information.
If your course has enabled Microsoft Office 365, you can upload a file from your Microsoft OneDrive for an assignment.
Like other file upload submissions, files uploaded from Office 365 are uploaded into your Canvas user files submissions folder.
Canvas accepts Microsoft Word, Microsoft PowerPoint, Microsoft Excel, and PDF types.
Notes:
If the Office 365 tab is not available in your submission, your institution has not enabled this feature.
Canvas will require you to authorize access to your OneDrive account.
In Office 365 assignments, you can only submit one file for your submission.
When Office 365 files are uploaded as a submission, later changes made to the file in OneDrive will not be updated in the submission.
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This document provides an overview of different methods for solving quadratic equations, including factoring, graphing, using the quadratic formula, and more. It begins by defining the general and standard forms of quadratic equations. It then explains how to write equations in standard form and discusses concepts like the vertex, completing the square, determining zeroes/roots, and the discriminant. Finally, it reviews several methods that can be used to find roots of quadratic and higher-order polynomial equations, such as factoring, graphing, the quadratic formula, synthetic division, and the remainder/factor theorems.
Graph Period 2
This document discusses graphing linear equations in one variable by plotting points and sketching vertical and horizontal lines. It explains that vertical lines are equations with only an x-variable (like x=4) and horizontal lines are equations with only a y-variable (like y=-1). The document provides examples of sketching the graphs of vertical and horizontal lines and finding their points of intersection. It concludes with examples of writing equations for vertical and horizontal lines that pass through given points.
คาบ 2
This document discusses graphing linear equations in one variable by plotting points and sketching vertical and horizontal lines. It explains that vertical lines are equations with only an x-variable (like x=4) and horizontal lines are equations with only a y-variable (like y=-1). The document provides examples of sketching the graphs of vertical and horizontal lines and finding their points of intersection. It concludes with examples of writing equations for vertical and horizontal lines that pass through given points.
2.3 graphs of functions
The document describes how to evaluate functions from graphs and algebraic expressions. It contains examples of functions f(x), g(t), h(v), k(p), g(r), and z(w) defined on axes or algebraically. The document provides examples of evaluating expressions like f(-2), g(0), h(1) using the different functions defined. It asks the reader to evaluate additional expressions using the given functions.
Calc 3.6a
This document provides guidance on sketching graphs based on analyzing key characteristics like intercepts, symmetry, domain, range, continuity, asymptotes, differentiability, extrema, and concavity. It works through examples of sketching rational functions and radical functions. For each example, it lists the first and second derivatives, intercepts, asymptotes, critical numbers, points of inflection, and symmetry. It then analyzes the characteristics of the graph over different parts of the domain based on the signs of the first and second derivatives. The goal is to use this calculus analysis to accurately sketch and label the graph with important points and features.
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solving graph of rational function using holes, vertical asymptote
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1150 day 11
Ratios, proportions, and percents can be used to compare quantities and solve problems. A ratio compares two quantities, like the number of squares to circles. Equal ratios form a proportion, which can be used to solve for unknown values. Percents represent a number out of 100. Common percent calculations include finding a percent of a number and converting between fractions, decimals, and percents. The percent proportion states that the percent equals the part divided by the whole quantity.
1150 day 10
This document covers topics related to repeating decimals, irrational numbers, and square roots. It includes examples of writing repeating decimals as fractions, ordering decimals, finding square roots, and using the Pythagorean theorem. Various math problems are presented along with step-by-step solutions to illustrate these concepts.
1150 day 9
1) Decimals represent numbers using place value with decimal points. To write decimals in expanded form, they are broken into place value terms using positive and negative exponents.
2) Fractions can be converted to decimals using division or by writing the fraction as a ratio of two numbers and setting up a proportion to solve for the decimal.
3) Scientific notation is used to write very large or small numbers in a standard form, such as 3.603 x 107. Operations can be done on decimals by lining up the decimal points or moving them with multiplication/division.
1150 day 8
This document provides an overview of rational numbers including:
- Integers and fractions written in the form a/b where a and b are integers and b ≠ 0
- Equivalent fractions represented by the same number
- Ordering and comparing rational numbers
- Converting between improper fractions and mixed numbers
- Basic operations of addition, subtraction, multiplication, and division of rational numbers
- Word problems involving rational numbers
1150 day 7
This document discusses prime and composite numbers, greatest common divisors (GCD), and least common multiples (LCM). It provides examples of finding the GCD and LCM of various numbers using different methods like the intersection of sets method, prime factorization method, and Euclidean algorithm. Key definitions include: a prime number has exactly two distinct positive divisors, a composite number has factors other than itself and 1, the GCD is the largest integer that divides numbers, and the LCM is the smallest number that is a multiple of the given numbers.
1150 day 6
Integers include positive and negative whole numbers. Absolute value is the distance from zero. Addition and subtraction on a number line involve moving left or right. Multiplication follows patterns based on sign. Division is the inverse of multiplication regarding sign. A number is divisible by another if it can be written as a product of an integer multiple. Divisibility tests identify patterns in a number's digits.
1150 day 5
Algebraic thinking involves recognizing patterns, modeling situations with symbols, and analyzing change. It relies on understanding variables to represent unknown quantities. The document traces the evolution of algebraic thinking from simple equations to more complex concepts like functions, composite functions, and properties of equations. It provides examples of how algebraic reasoning and symbols can be used to represent and solve real-world problems.
1150 day 4
The document discusses various aspects of whole number operations including:
1) Place value systems and how numbers are represented in bases other than 10 such as base 5 and base 12.
2) Algorithms for addition, subtraction, multiplication, and division using various models and representations.
3) Properties of operations like the commutative, associative, identity, and zero properties of multiplication.
1150 day 3
The document discusses different number systems used throughout history including Hindu-Arabic, Egyptian, Babylonian, Mayan, and Roman systems. It also covers basic concepts of whole numbers such as addition, subtraction, and their properties. Different models are presented to demonstrate whole number operations including set, number line, take-away, comparison, and missing addend models.
1150 day 2
The document discusses sets and set operations including defining sets, elements, cardinal and ordinal numbers, equal and equivalent sets, subsets, Venn diagrams, and set operations like union, intersection, and complement. Examples are provided to illustrate concepts like finite and infinite sets, subsets, Venn diagrams representing multiple sets and operations, and using Venn diagrams to solve problems involving sets and their relationships.
1150 day 1
The number of terms in each sequence can be determined by finding a one-to-one correspondence with the counting numbers.
Alg2 lesson 13-5
This document discusses solving an ambiguous triangle where it is not a right triangle and the measures of angles and sides are not fully known. It determines that trigonometric functions and the Law of Sines cannot be used. It identifies that the Law of Cosines is the appropriate theorem to use to begin solving since it requires only knowing the measures of three sides.
Alg2 lesson 13-4
The document discusses solving for unknown sides and angles of triangles using trigonometric functions like the law of sines. Several multi-step example problems are worked through that involve determining if a triangle is possible based on given information, finding missing side lengths, and calculating angles. Diagrams are included to illustrate each step of the example problems.
Alg2 lesson 13-3
The document contains instructions for finding trigonometric function values given specific angle measurements or terminal side locations. It includes:
1) Finding the six trig functions of an angle with a terminal side at (8, -15)
2) Finding the six trig functions of an angle with a terminal side at (-3, 4)
3) Finding the reference angle of 330°
4) Finding the reference angle of an angle in Quadrant III
5) Finding the value of sin(135°)
6) Finding the remaining five trig functions of an angle in Quadrant III with a terminal side of (-4, -3)
Alg2 lesson 13-3
The document contains instructions for finding trigonometric function values given specific angle measurements or terminal side locations. It includes:
1) Finding the six trig functions of an angle with a terminal side at (8, -15)
2) Finding the six trig functions of an angle with a terminal side at (-3, 4)
3) Finding the reference angle of 330°
4) Finding the reference angle of an angle in Quadrant III
5) Finding the value of sin(135°)
6) Finding the remaining five trig functions of an angle in Quadrant III with a terminal side of (-4, -3)
Alg2 lesson 13-2
The document describes how to draw angles in standard position on a unit circle and convert between degrees and radians. It provides examples of drawing angles such as 210°, -45°, and 540° in standard position and rewriting angles such as 30°, 45°, and an unspecified angle in radians and degrees. It also gives examples of finding coterminal angles for 210° with one positive and one negative measure.
Alg2 lesson 13-1
This document discusses trigonometric functions and their ratios, including sine, cosine, tangent, cosecant, secant, and cotangent. It provides examples of using trigonometric functions to solve values in right triangles, including finding missing side lengths and angle measures. Special right triangles with ratios of 1/2, 1/√3, and 1/√2 are also covered.
Alg2 lesson 11-2 and 11-4
The document provides information about arithmetic and geometric series. It defines arithmetic and geometric series, provides examples of finding sums of arithmetic series using formulas, and defines the key terms (first term, common ratio, number of terms, last term) used in the formula to calculate the sum of a geometric series.
Alg2 lesson 11-3
The document discusses geometric sequences, which are patterns of numbers where each term is found by multiplying the previous term by a constant called the common ratio. It provides examples of geometric sequences with different common ratios and shows how to write equations to find the nth term in a sequence. It also explains how to find specific terms in a geometric sequence and calculates the geometric means between two nonconsecutive terms.
Alg2 lesson 11-1
The document discusses arithmetic sequences, which are patterns of numbers where each term is found by adding a constant value to the previous term. It provides examples of arithmetic sequences with different common differences and shows how to write equations to determine the nth term. It also explains how to find terms within an arithmetic sequence and determine arithmetic means between two non-consecutive terms.
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Alg2 Lesson 9 3
- 1. Vertical Asymptote Horizontal Asymptote x = 4 y = 1 Hole At x = -2
- 4. Example 3-1a Undefined for x = –2 and –3 Hole at x = -2 Vertical asymptote is the line x = -3 Horizontal asymptote: the line y = 1 Determine the equations of any asymptotes and the values of x for any holes in the graph of
- 5. Graph VA: x = -3 HA: y = 1 Hole at x = -2 -2 -2 = -4 (-2, -4)
- 6. Graph x -6 -5 -4 -1 0 1 -8/-3 or 8/3 -7/-2 or 7/2 -6/-1 or 6 -3/2 -2/3 -1/4
- 7. Example 3-2a VA: x = –1 No holes HA: y = 1 Graph
- 8. Example 3-2b x f ( x ) – 4 -4/-3 = 4/3 – 3 -3/-2 = 3/2 – 2 -2/-1 = 2 0 0/1 = 0 1 1/2 2 2/3 3 3/4 Graph
- 9. Example 3-3a Undefined for x = 2 Hole at x = 2 2 (2, 4) x -1 0 1 1 2 3 Graph