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This document provides an overview of key concepts in functions and their graphs including: - Properties of lines like slope and equations of lines - Basic functions including domain, range, and the vertical line test - Transformations of graphs including translations, reflections, stretches, and shrinks - Combining functions using addition, subtraction, multiplication, and division - Graphing polynomials, rational functions, and analyzing their properties - Exponential and logarithmic functions and their properties - Graphing parabolas, ellipses, and hyperbolas and analyzing their equations
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1. The document discusses different families of lines graphed on a number plane. The red lines are parallel and have negative x terms in their equations. The blue lines are also parallel and have positive 2x terms.
2. Three lines intersect at the origin - the green line y=x, the blue line y=2x, and the red line y=-x. These lines all lack a constant term.
3. Investigating the graphs shows that lines with a negative x term slope down as x increases, while lines with a positive x term slope up. The constant b relates to where the line intercepts the y-axis, with higher b above the axis and lower b below it.
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- To graph a linear inequality, first solve it for y and graph the resulting equation as a line. Then, test a point not on the line to determine which side of the line to shade based on whether it satisfies the inequality.
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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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- A linear inequality describes a region of the coordinate plane bounded by a line. Any point in the shaded region is a solution to the inequality.
- To graph a linear inequality, first solve it for y and graph the resulting equation as a line. Then, test a point not on the line to determine which side of the line to shade based on whether it satisfies the inequality.
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Okay, let's solve this step-by-step:
1) FOIL the expression: y = 3(x + 2)(x + 2) - 4
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= 3x^2 + 12x + 12 - 4
= 3x^2 + 12x + 8
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Where: a = 3
h = 0
k = 8
So the equation in vertex form is:
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y = 3x^2 + 8
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A polynomial function is a function where all exponents are non-negative integers. It is of the form f(x) = anxn + an-1xn-1 + ... + a1x + a0, where n is a non-negative integer and the coefficients ai are real numbers. The degree of a polynomial is the highest exponent present, and the leading coefficient is the coefficient of the term with the highest degree. The end behavior of a polynomial function can be determined by looking at the sign of the leading coefficient and whether the highest degree is even or odd. A polynomial function has at most n real zeros, where n is its degree, and at most n-1 turning points.
2.1 Functions and Their Graphs
The domain is the set of all possible square footages, which is 1450 to 2100 square feet. The range is the set of all possible costs, which would be $75 * 1450 = $108,750 to $75 * 2100 = $157,500.
So the domain is [1450, 2100] and the range is [108750, 157500].
Functions
1. The document discusses functions and their graphical representations. It defines key terms like domain, range, independent and dependent variables.
2. It provides examples of different types of functions including constant, linear, quadratic, and rational functions. Linear functions have a straight line graph, quadratic functions have a parabolic graph, and rational functions have a hyperbolic graph.
3. The document also discusses how to graph systems of equations, which can have a single solution point (consistent independent), infinite solutions along the same line (consistent dependent), or no solution (inconsistent parallel lines).
4.2 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 = -2
k = 8
So the equation in vertex form is:
y = 3(x - (-2))^2 + 8
y = 3(x + 2)^2 - 4
General Mathematics - Representation and Types of Functions
It is a powerpoint presentation that will help the students to enrich their knowledge about Senior High School subject of General Mathematics. It is comprised about the representation, definition, and types of functions.
Advanced algebra (some terminologies)
The document defines and describes key concepts related to rectangular coordinate systems and functions. It introduces the x-axis, y-axis, and origin that make up the rectangular coordinate system. It then defines various types of functions like linear, quadratic, absolute value and their graphs. Key characteristics of functions like domain, range, and intercepts are also summarized.
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This document provides an overview of different types of functions that may be assessed on the ACT exam, including linear, quadratic, trigonometric, logarithmic, and exponential functions. It discusses key concepts for each type of function such as domain and range, asymptotes, amplitude and period for trig functions, and properties of logarithmic and exponential growth/decay functions. Examples are provided to illustrate how to work with each function type, including solving equations, finding features of graphs, and recognizing standard forms.
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This document provides an overview of different types of functions that will be investigated in an Algebra II course. It discusses linear, quadratic, cubic, exponential, and absolute value functions. For each function type, it describes their graphical representation and symbolic notation. Students are instructed to graph examples of each function type using graph paper and calculators to check their work. The document emphasizes that students will explore these and new functions in more depth during the Algebra II course.
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This document provides an overview of linear and non-linear functions for an 8th grade math class. It defines linear functions as those that form a straight line when graphed, while non-linear functions do not. Linear functions can be represented by graphs, equations in slope-intercept form (y=mx+b), or tables with constant rates of change for both x- and y-values. Non-linear functions have curved graphs or tables without constant rates of change. Examples of linear and non-linear equations and tables are provided to illustrate the differences.
Lecture 01-2 (Functions).pptx
This document provides information about an Applied Calculus course taught by Imran Qasim at Mehran University of Engineering and Technology. The key points are:
1) The course covers topics in differential and integral calculus, including functions, limits, derivatives, integrals, and their applications.
2) Students are expected to have prior knowledge of functions, limits, and differentiation before taking the course.
3) The course contents will help students develop expertise in techniques for differentiation and integration, as well as apply calculus to solve real-world problems.
Ap calculus warm up 8.27.13
To find the limit of a function as x approaches positive or negative infinity, examine the degrees of the numerator and denominator polynomials. If the degrees are equal, the limit is the ratio of the leading coefficients; if the numerator degree is greater, the limit does not exist; if the denominator degree is greater, the limit is 0. Rational functions can have horizontal or vertical asymptotes. Examples show evaluating limits of rational functions by comparing degrees and leading coefficients as x approaches infinity. Practice problems apply these techniques to additional rational function limits.
Chap7 2 Ecc Intro
Elliptic curve cryptography uses elliptic curves over finite fields for public-key encryption. It offers the same security as other public-key cryptosystems using smaller key sizes. The points on an elliptic curve over a finite field form a finite abelian group which can be used for cryptographic operations like point addition. Point addition involves calculating the slope between two points and using it to find the x-coordinate of the sum point, while point doubling uses the tangent line to find the double of a point.
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Linear regression is a supervised learning technique used to predict continuous valued outputs. It fits a linear equation to the training data to find the relationship between independent variables (x) and the dependent variable (y).
Gradient descent is an algorithm used to minimize the cost function in linear regression. It works by iteratively updating the parameters (θ values) in the direction of the steepest descent as calculated by the partial derivatives of the cost function with respect to each parameter. The learning rate (α) controls the step size in each iteration.
Multivariate linear regression extends the technique to problems with multiple independent variables by representing the hypothesis and parameters as vectors and matrices, allowing gradient descent to optimize the parameters to fit the linear model
4.2 vertex and intercept form
This document discusses graphing quadratic functions in vertex and intercept forms. It defines key terms like vertex, axis of symmetry, and parabola. It explains how to graph quadratic functions written in vertex form and intercept form. Examples are provided for graphing functions in each form. The document also discusses how to change between the vertex, intercept, and standard forms of a quadratic equation.
PP1 Presentation
This document provides an overview of functions and their representations. It discusses the domain and range of a function, and how functions can be represented verbally, visually, numerically, and algebraically. It also describes the vertical line test used to determine if a graph represents a function. Inverse functions and the horizontal line test are introduced. Common types of functions covered include linear, polynomial, cubic, rational, and trigonometric functions. Mathematical modeling using functions is also briefly discussed.
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Cochino’s math
- 1. Cochino’s Math By Jocelyn Hernandez, Leslie Zamudio, Horacio Sevilla & Yesenia Gonzalez
- 2. Functions and Their Graphs • Properties of Lines • Slope m=y2-y1/x2-x1, • Point slope y-y1=m(x-x1), Slope Intercept y=mx+b • Vertical X=Y Horizontal Y=X
- 3. Basic Functions, Functions and Graphs • Function: A function has a value in the domain with exactly one value in the range • Domain: The input or (x) • Range: The output or (y) • Vertical line test: if it passes it’s a function • Solving for a function: Algebraic, Numeric or Graphically • Continuity of a graph: Contiguous • Increasing and decreasing graphs. • Asymptotes: a line that a graph approaches but never reaches
- 4. Transformations (Shifts, Stretches, and Reflections) • Horizontal Translations: • Y=f(X-C) Translation to the right by C units • Y=f(X+C) Translation to the left by C units • Vertical Translations: • Y=F(X)+C Translation up by C units • Y=F(X)-C Translation down by C units • Reflections across the X-axis: y=-f(x) • Reflections across the Y-axis= f(-x)
- 5. Transformations (Shifts, Stretches, and Reflections) • Stretches and Shrinks: • Horizontal: Y=F(X/C) {A stretch by a factor of C if C>1} • {a shrink by a factor of C if C<1} • Vertical: Y=C*F(X) {A stretch by a factor of C if C>1} • {A shrink by a factor of C if C<1}
- 6. Combination of Functions • Sum: (F+G)=F(X)+G(X) • Difference: (F-G)(X)=F(X)-G(X) • Product: (FG)(X)=F(X)*G(X) • Quotient: (F/G)(X)=F(X)/G(X), Provided G(X) cannot equal 0
- 7. Polynomials and Rational Functions • Ways to solve a Quadratic Equation: Factoring, Using the Quadratic Formula, Completing the Square ax2 + bx = c • Polynomial Function • One-To-One Functions • Horizontal Line Test -If some horizontal line intersects the graph of the function more than once,then the function is not one-to-one. -If no horizontal line intersects the graph of the function more than once,then the function is one-to-one.
- 8. Polynomials and Rational Functions • Synthetic Division: • Real zeros and complex numbers -complex numbers are numbers such as : 4+3i , 5i+i etc... • real zeros are the intercepts of a quadratic equation
- 9. Polynomials and Rational Functions • Rational Functions :To graph a rational function, you find the asymptotes and the intercepts, plot a few points, and then sketch in the graph example equation:
- 10. Polynomials and Rational Functions • Fundamental Theorem of Algebra • Any polynomial of degree n ... has n roots, but you may need to use complex numbers example of a polynomial • this one has 3 terms • The Degree of a Polynomial with one variable is the largest exponent of that variable.
- 13. Exponential and Logarithmic Functions • Exponential Function : f(x)=a. bx • Logarithmic Function: y= Logbx • Properties of Logs: • Product: logbMN=logbM+logbN • Quotient: LogbMN=logbM-LogbN • Power: LogbNy=ylogbN
- 14. Exponential and Logarithmic Functions • Basic Common Logarithms Functions: • -Log101=0 because 100=1 • -Log1010=1 because 101=10 • -Log1010y=y because 10y=10y • -Loglogx=x because Logx=logx • Log always finds the exponent!!
- 16. Ellipses with center (h,k)
- 17. Hyperbolas