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The document discusses various types of transformations that can be performed on the graph of the function f(x)=logax. It explains that dilations on the x-axis are shown by coefficients in front of x in the equation, dilations on the y-axis by coefficients in front of the log term. Translations are shown by adding or subtracting terms from x or at the end of the equation for x-axis or y-axis translations respectively. Reflections are shown by changing the sign in front of terms or dividing the entire equation by 1. It also discusses finding degrees from radians and vice versa, finding intercepts of log equations, and determining the composition of two functions.
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= 3x^2 + 12x + 8
2) This is in the vertex form: y = a(x - h)^2 + k
Where:
a = 3
h = -2
k = 8
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y = 3(x - (-2))^2 + 8
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This document contains notes and formulas for SPM Mathematics for Forms 1-4. It covers topics such as solid geometry, circle theorems, polygons, factorisation, expansion of algebraic expressions, indices, algebraic fractions, linear equations, simultaneous linear equations, algebraic formulas, linear inequalities, statistics, quadratic expressions and equations, sets, mathematical reasoning, the straight line, trigonometry, angle of elevation and depression, lines and planes. Formulas and properties are provided for calculating areas and volumes of solids, solving different types of equations, and relationships in geometry, trigonometry and statistics. Examples are included to demonstrate solving problems and using the various formulas and concepts.Elementary algebra notes 001.pdf
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This document is the preface to the 2012-2013 second edition of an elementary algebra textbook published by the Department of Mathematics at College of the Redwoods. It contains copyright information, noting that the content is available for free use in educational institutions but is not in the public domain. The preface also includes a table of contents that lists the chapters and their section topics covered in the textbook.Linear Equations And Graphing
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Here are the steps to solve 2(x 3) using algebra tiles:
Draw x and 3 tiles. Then place 2 sets of those tiles side by side.
x 3 x 3
Group the like tiles.
2x 6
So, 2(x 3) = 2x 6
The distributive property allows you to multiply a single term outside parentheses over each term inside. This is the same process as using algebra tiles to model the multiplication.
1. Use the distributive property to write each expression as a single term.
a) 4(x + 3) = _____________________ b) 3(2xLesson 3.4
Lesson 3.4guest78a909
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This document discusses transformations of graphs, including shifting graphs horizontally and vertically, changing the amplitude to model changes in loudness, compressing graphs to fit standard view windows, and flipping graphs across the x-axis or y-axis. Some key points:
- Shifting a graph horizontally is done by adding or subtracting a value from the input, like f(x-2) or f(x)+3.
- Shifting vertically is done by adding or subtracting from the entire function, like f(x)+2.
- Increasing amplitude stretches a graph vertically, modeling increased loudness. Decreasing amplitude compresses vertically.
- Compressing a graph fits it within the standard view by multiplying theForm 5 Additional Maths Note
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This document contains notes on additional mathematics including topics on progression, linear laws, integration, and vectors. Some key points:
- It discusses arithmetic and geometric progressions, defining the terms and formulas for finding terms and sums. Examples are worked through finding terms, sums, and differences between sums.
- Linear laws are explained including lines of best fit, converting between linear and non-linear forms using logarithms, and working through examples of finding equations from graphs.
- Integration techniques are outlined including formulas for integrals of powers, areas under and between curves, volumes of revolution, and the basic rules of integration. Worked examples find areas and volumes.
- Vectors are introduced including addition using the triangle graphs of functions 2
graphs of functions 2larasati06
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Here are the steps to solve this problem:
1) The given equation is: y = x2 + 2x
2) To complete the table, we need to calculate the value of y when x = -3 and when x = 1
3) When x = -3:
y = (-3)2 + 2(-3)
y = 9 - 6
y = 3
4) When x = 1:
y = 12 + 2(1)
y = 1 + 2
y = 3
So the completed table is:
Table 1
x -3 1
y 3 3
(b) Sketch the graph of y = x2 + 2Project in Mathematics. Functions
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This document is a student's math project submitted to their teacher. It covers several topics in math including:
1) Evaluating functions by substituting values into equations. Examples are worked out for different functions.
2) Inverse functions and how to think of a function transforming one value into another.
3) Synthetic division, working through an example problem.
4) Exponential functions, working through examples of exponential equations.
5) Logarithm functions, explaining how to arrange logarithm equations before solving them and working through examples.
6) Converting between degrees and radians, working through examples of both degree to radian and radian to degree conversions.KSSM Form 4 Additional Mathematics Notes (Chapter 1-5)
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Sam maths pres
- 1. Transforming GraphsYear 12 Mathematical MethodsSam Roberts
- 2. Graphing transformations of the graph of f(x)=logaxDilation on the x-axis is shown by the number at the front of the equation. E.g. f(x)=3loga xDilation on the y-axis is shown by the number in front of the pronumeral. E.g. f(x0=loga 2x
- 3. Graphing transformations of the graph of f(x)=logaxTranslations on the x-axis is shown by the number added or subtracted to x. E.g. f(x)=loga (x-5)Translation on the y-axis is shown by the number at the end of the equation. E.g. f(x0=loga x +5
- 4. Graphing transformations of the graph of f(x)=logaxReflection on the x-axis is shown by the symbol at the front of the equation. E.g. f(x)=-loga xReflection on the y-axis is shown by dividing the equation to 1. E.g. f(x)= 1 _loga x
- 5. Graphing circular function transformations
- 6. Finding the degrees when the result is in radians. Vice Versa.Eg3 x 180 = 540 = 135属 4 4Eg 30属 x = 30 = 180 180 6
- 8. Finding axis interceptsLet y = 0 to find x intercepts
- 9. Finding the inverse of log equationsSub x as y to find the inverseLogaa烹=x for all x Rlogaxa =x for x R
- 10. Finding a rule for fg(x)Sub the equation of g(x) where the x in f(x) equationEg. g(x)=x and f(x)fg(x)=sin(x)