Logsexponents
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Captain Jabbamathee's steps to graph the natural logarithm function y = |log x| are: 1. Graph the inverse function y = 5^x to determine the asymptote and domain and range. 2. Graph y = log x by reversing the x and y values of the inverse graph y = 5^x. 3. Make all values positive to account for the absolute value. The graph has an asymptote of x = 0, domain of (0, ∞), range of [0, ∞), and increases and decreases.
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linear, quadratic, cubic, power, trigonometric, exponential, and logarithmic functions are discussed.
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The document discusses transformations of linear functions. It provides examples of translating graphs vertically and horizontally by adding or subtracting values from inputs and outputs. It also discusses reflecting graphs over the x-axis or y-axis by multiplying inputs or outputs by -1. Horizontal and vertical stretches and shrinks are described as multiplying the inputs or outputs by a scale factor. The key is that transformations preserve the shape of the graph but can change its position, orientation, or scale.
1. Graph the exponential function by hand. Identify any asymptotes.docx
1. Graph the exponential function by hand. Identify any asymptotes and intercepts and determine whether the graph of the function is increasing or decreasing. (Enter NONE in any unused answer blanks.)
Equation of horizontal asymptote:
Equation of vertical asymptote:
Value of y-intercept
Value of x-intercept
The function is
2. Use the graph of y = 2x to match the function with its graph.
A
B
C
D
y = 2x – 4
y = 2x – 5
y = 2x + 4
y = 2–x
3. Use the graph of f to describe the transformation that yields the graph of g. Then sketch the graphs of f and g by hand.
f(x) = −2x, g(x) = 5 − 2x
The graph of g(x) = 5 − 2x is a vertical shift five units downward of f(x) = −2x.
The graph of g(x) = 5 − 2x is a horizontal shift five units to the left of f(x) = −2x.
The graph of g(x) = 5 − 2x is a vertical shift five units upward of f(x) = −2x.
The graph of g(x) = 5 − 2x is a horizontal shift five units to the right of f(x) = −2x.
Sketch the graphs of f and g.
4. Use a graphing utility to construct a table of values for the function. (Round your answers to three decimal places.)
x
f(x) = 5x − 3
-1
0
1
2
3
Sketch the graph of the function.
Identify any asymptotes of the graph. (Enter NONE in any unused answer blanks.)
vertical asymptote
x
=
horizontal asymptote
y
=
5. Use a graphing utility to construct a table of values for the function. (Round your answers to three decimal places.)
x
g(x) = 4 − e−3x
-4
-3
-2
-1
0
Sketch the graph of the function.
Identify any asymptotes of the graph. (Enter NONE in any unused answer blanks.)
vertical asymptote
x
=
horizontal asymptote
y
=
6. Fill in the blank.
If
x = ey,
then y = .
7. For what value of x is
ln x = ln 9?
x =
8. Write the logarithmic equation in exponential form. For example, the exponential form of
log5 25 = 2 is 52 = 25.
log2 512 = 9
=
9. Write the logarithmic equation in exponential form. For example, the exponential form of log5 25 = 2 is 52 = 25.
log
1
100,000,000
= -8
=
10. Write the exponential equation in logarithmic form. For example, the logarithmic form of
23 = 8 is log2(8) = 3.
43/2 = 8
log
=
11. Use a calculator to evaluate the function at the indicated value of x. Round your result to three decimal places.
f(x) = log10(x) x = 4/5
12. Solve the equation for x.
log10(102) = x
13. Write the logarithmic equation in exponential form. For example, the exponential form of ln(5) = 1.6094... is e1.6094... = 5. (Do not use ... in your answer.)
=
14. Write the exponential equation in logarithmic form. For example, the logarithmic form of
e2 = 7.3890 is ln 7.3890 = 2.
(Do not use ... in your answer.)
e2.2 = 9.0250
ln
=
15. Use the properties of natural logarithms to rewrite the expression.
5 ln(e5)
16. Use the properties of logarithms to rewrite and simplify the logarithmic expression.
ln
9
e9
17. Use the properties of logarithms to expand the expression as a ...
C:\Fakepath\Stew Cal4e 1 6
The document discusses inverse functions and logarithms. It begins by introducing the concept of an inverse function using an example of bacteria population growth over time. It then defines inverse functions formally and discusses their key properties. The document explains that a function must be one-to-one to have an inverse function. It introduces the natural logarithm as the inverse of the exponential function with base e and discusses properties of logarithmic functions like logarithmic laws. Graphs of exponential, logarithmic and natural logarithmic functions are presented.
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.
Algebra 2. 9.15. Intro to quadratics
The document provides information about graphing and transforming quadratic functions:
- It discusses graphing quadratic functions by making tables of x- and y-values and plotting points. Examples show graphing f(x) = x^2 - 4x + 3 and g(x) = -x^2 + 6x - 8.
- Transformations of quadratic functions are described as translating the graph left/right or up/down, reflecting across an axis, or stretching/compressing vertically or horizontally. Examples demonstrate translating, reflecting, and compressing the graph of f(x) = x^2.
- The vertex form of a quadratic function f(x) = a(x-h)^2 +
Algebra 2. 9.16 Quadratics 2
This document provides examples and explanations for identifying key properties of quadratic functions presented in standard form, including:
1. Determining whether the graph opens upward or downward based on the sign of the leading coefficient a.
2. Finding the axis of symmetry by setting b/2a equal to x.
3. Finding the vertex by substituting the x-value from the axis of symmetry into the original function to determine the y-value.
4. Identifying the y-intercept from the constant term c.
Steps are demonstrated for graphing quadratic functions based on these properties, finding minimum/maximum values, and stating the domain and range. Examples analyze functions algebraically and graphically.
Functions
The document discusses functions and evaluating functions. It provides examples of determining if a given equation is a function using the vertical line test and evaluating functions by substituting values into the function equation. It also includes examples of evaluating composite functions using flow diagrams to illustrate the steps of evaluating each individual function.
INTRODUCTION TO MATLAB presentation.pptx
Here are the steps to solve this problem numerically in MATLAB:
1. Define the 2nd order ODE for the pendulum as two first order equations:
y1' = y2
y2' = -sin(y1)
2. Create an M-file function pendulum.m that returns the right hand side:
function dy = pendulum(t,y)
dy = [y(2); -sin(y(1))];
end
3. Use an ODE solver like ode45 to integrate from t=0 to t=6pi with initial conditions y1(0)=pi, y2(0)=0:
[t,y] = ode45
April 13, 2015
This document contains information about a math class that is reviewing quadratic functions. It includes:
1. An outline of the class agenda which focuses on reviewing key concepts like how the b-value affects the parabola and completing classwork.
2. Details about grading which includes assignments, homework, tests, the final exam, and notebook checks.
3. Sample problems and class notes focused on quadratic functions, including the axis of symmetry, vertex, graphing techniques, and how changing a, b, and c values impacts the parabola.
4. Examples of completing the steps to graph quadratic functions like plotting points and reflecting over the axis of symmetry.
How to graph Functions
This lesson discusses graphing functions. It begins with examples of graphing functions given a limited domain by finding ordered pairs that satisfy the function and plotting the points. It then covers graphing functions with a domain of all real numbers by choosing values of x, finding corresponding y-values, plotting points to see a pattern, and drawing a line with arrowheads. The lesson shows how to use graphs to find function values and solve real-world problems by limiting domains to non-negative values.
4 4 graphingfx
This lesson discusses graphing functions. It begins with examples of graphing functions given a limited domain by finding the y-values that correspond to the given x-values. It then covers graphing functions with an unlimited domain by choosing sample x-values and connecting the points with a line or curve. The document provides examples of interpreting graphs to find function values and solving real-world word problems by graphing and interpreting functions.
Gr 10 absolute value functions and graphs
The document discusses absolute value functions and graphs. It defines the vertex as the maximum or minimum of the graph. Examples are given to identify the vertex of different absolute value functions and how to graph them by making a table. The key aspects are to find the vertex, make a table of values with at least 5 points, and graph the function by connecting the points. Similarities and differences between graphs are noted. Additional examples graph different absolute value functions.
Exponents and Logs
This document covers exponents, logarithms, and their functions. It defines exponential and logarithmic expressions, reviews rules for manipulating exponents and logarithms, and examines the graphs of exponential and logarithmic functions. It provides examples of applying exponent and logarithm rules, and contains activities involving evaluating exponential and logarithmic expressions and graphing their related functions.
Deriving the inverse of a function1
The document explains how to derive the inverse of a function. It demonstrates finding the inverse of functions f(x)=2x+5, g(x)=x/7+4, and h(x)=(5x+4)/(x+9) through the following steps: 1) rewrite the original function as y=fx(x), 2) interchange x and y, 3) isolate y to obtain the inverse function f^-1(x). The inverse of f(x) is f^-1(x)=(x-5)/2. The inverse of g(x) is g^-1(x)=7(x-4). The inverse of h(x) is h^-1(x)=(
D.e.v
The document summarizes how various Avengers calculate and solve different math problems related to their superpowers and equipment. Captain America finds the area of his shield as a function of circumference. Iron Man solves a quadratic equation to power his arc reactor. The Hulk smashes a wall and calculates how many pieces it breaks into. Hawkeye finds the equation of a parabola based on its vertex and a point. Thor analyzes the domain and graph of a skipping hammer's movement. Black Widow determines the domain and range of a radical function.
Functions & graphs
1) The document discusses various topics in mathematics including sets, functions, composite functions, exponential and logarithmic graphs, and graph transformations.
2) It provides definitions and examples of sets, functions, and how to represent functions using formulas, arrow diagrams, and graphs. Composite functions are defined as functions of other functions.
3) The document explains how to graph exponential and logarithmic functions and describes the key features of these graphs. It also discusses how different transformations can move a graph in various ways, such as reflecting it or stretching/squashing it.
Deriving the inverse of a function2 (composite functions)
The document discusses deriving the inverse of a composite function. It provides examples of composite functions and their inverses. Specifically:
- It defines a composite function as first applying one function, then another, denoted fg. The inverse of a composite function is found by first deriving the composite function, then taking its inverse.
- An example problem finds the inverses of (fg), (gf), and (hg), where f, g, and h are functions of x. The composite functions are derived then their inverses are obtained by swapping x and y.
- The inverse of (fg) is (3x+16)/3. The inverse of (gf) is (3x+
Transformations
The document discusses how to graph functions by applying transformations to basic graphs. These transformations include vertical and horizontal shifts which move the graph up/down or left/right, reflections which flip the graph across an axis, and stretching or shrinking which make the graph taller or narrower. By identifying the transformations in an equation, one can graph it by applying the corresponding operations to the original function graph.
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1. Graph the exponential function by hand. Identify any asymptotes.docx
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Credits
The document appears to be a fictional story told in 6 chapters that incorporates photos from Flickr to illustrate various concepts related to math, school, homework, and conic sections. Photos include images of studying, soccer, bees, homework, and geometric shapes like ellipses. The story seems to follow a student learning about different math concepts both in and out of school.
Conics
The document discusses graphing ellipses. It appears to be a photo of a whiteboard with an equation for an ellipse written on it along with some steps to graph the ellipse. The key information provided is an equation for an ellipse and a process to graph the shape defined by that equation.
Logproblempres
The document discusses exponential modeling and its applications to bee population growth. It provides two cases for using exponential functions, including an example of calculating future bee population given an original population, growth rate, and time. A second example solves for the growth rate given initial population and future population after a set time. Exponential functions can model a variety of growth patterns in nature, business, and science.
Transformationss
The document discusses solving homework problems involving graph transformations. It explains how to graph the absolute value of a function f(x) shifted down by 2 units. It then asks to sketch the function y = -2f(1/2x – 2) + 3, identifying that the transformations occur in this order: 1) vertical stretch by factor of -2, 2) horizontal compression by factor of 1/2, 3) horizontal shift of -4, and 4) vertical shift of +3. The steps to solve these graph transformation problems are demonstrated.
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The document discusses solving homework problems involving graph transformations. It explains how to graph the absolute value of a function f(x) shifted down by 2 units. It then asks the reader to sketch the function y = -2f(1/2x – 2) + 3, identifying that the transformations occur in this order: 1) vertical stretch by a factor of -2, 2) horizontal compression by a factor of 1/2, 3) horizontal shift of -4 units, and 4) vertical shift of +3 units. The resulting graph is shown with the transformed coordinates.
Trigidentities
The document discusses trigonometric identities and provides steps to solve related problems. It first reviews that sine and cosine are the basic trig functions and introduces the Pythagorean identities. It then presents two example problems and provides multi-step solutions using trig identities, including the Pythagorean identities, SOHCAHTOA, and "cosine dance" techniques. The goal is to simplify trigonometric expressions and solve for desired trig functions.
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Logsexponents
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