Alg1 ch1101example123
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The document provides examples and steps for graphing functions of the form y=ax, y=ax+k, and identifying their domains and ranges. It discusses vertical stretches, shrinks, translations, and reflections of the graph of y=x. Example 1 graphs y=3x, which is a vertical stretch of y=x by a factor of 3. Example 2 graphs y=-0.5x, a vertical shrink and reflection of y=x. Example 3 graphs y=x+2, a vertical translation of y=x up by 2 units. The guided practice graphs additional functions and identifies their transformations compared to y=x.
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The document discusses key concepts about function graphs including:
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- Notable function graphs include constant, sign, identity, and linear functions
- The graph of a quadratic function f(x)=ax^2+bx+c is a parabola that can be used to find the vertex and x-intercepts
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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.
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3. Using function mode, solving for x as a function of y, and using DrawInv to graph the inverse relation.
4. Using function mode and the Solve() command to numerically solve the implicit equation for y as a function of x.
5. Using polar mode by rewriting the implicit equation in terms of r and θ and graphing r
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The document discusses graphing quadratic functions. It begins with reviewing key concepts like the vertex and axis of symmetry. The effects of the a, b, and c coefficients on the parabola are explained. Examples are provided to show how changing these values affects the width, direction opened, and vertical translation of the graph. The class will graph various quadratic functions by finding the axis of symmetry, vertex, y-intercept, and other points to plot the parabola. Students are assigned class work problems to graph quadratic functions and show their work.
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This document provides an overview of various types of functions and their graphs. It begins with linear functions of the form y=mx+c and discusses how shifting these functions along the x- or y-axis changes their graphs. It then covers quadratic, square root, cube, reciprocal, constant, identity and absolute value functions. Piecewise, polynomial, algebraic, and transcendental functions are also defined. The document discusses bounded vs unbounded functions and concludes by examining circular and hyperbolic functions and their graphs.
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This document defines and provides examples of quadratic functions. A quadratic function is a polynomial function of degree two in the form f(x) = ax^2 + bx + c, where a ≠ 0. It must have the highest degree of 2, a non-zero coefficient for x^2, and no negative or rational exponents. Examples are provided to identify quadratic functions from sets of equations. The graph of a quadratic function is a parabola that can open upward or downward depending on the sign of a. Additional examples demonstrate how changing the coefficients affects the graph shape. The vertex form of a quadratic function is given as f(x) = a(x-h)^2 + k, where the vertex is (
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The document provides an agenda and lesson plan for an Algebra I class that includes the following:
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2. Students will then summarize their notes and have a teacher dialogue.
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Calculus and Numerical Method =_=
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This document discusses graphing rational functions. It defines key concepts like domain, range, intercepts, zeros, and asymptotes. An example rational function f(x)=x-2/(x+2) is used to demonstrate how to find these values and graph the function. The domain is all real numbers except -2, the x-intercept is 2, and the y-intercept is -1. The vertical asymptote is at x=-2. Horizontal asymptotes occur when the degree of the numerator is less than, equal to, or greater than the degree of the denominator.
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The document is about quadratic polynomial functions and contains the following information:
1. It discusses investigating relationships between numbers expressed in tables to represent them in the Cartesian plane, identifying patterns and creating conjectures to generalize and algebraically express this generalization, recognizing when this representation is a quadratic polynomial function of the type y = ax^2.
2. It provides examples of converting algebraic representations of quadratic polynomial functions into geometric representations in the Cartesian plane, distinguishing cases in which one variable is directly proportional to the square of the other, using or not using software or dynamic algebra and geometry applications.
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This document provides an overview of various mathematical models and functions, including:
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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.
Modeling with Quadratics
- The document discusses quadratic functions and their graphs. It explains that the graph of a quadratic function is a parabola, which is a U-shaped curve.
- It describes how to write quadratic functions in standard form and use that form to sketch the graph and find features like the vertex and axis of symmetry.
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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 ...
Function evaluation, termination, vertical line test etc
The document discusses functions and their inverses. It defines key terms like domain, range, and inverse functions. It provides examples of determining if a relationship is a function using the vertical line test or one-to-one mapping. The document also covers finding inverses of straight lines, parabolas, exponential graphs and logarithmic graphs. It shows how to sketch these graphs and determine their domains and ranges. Finally, it discusses solving cubic expressions through finding factors and remainders.
Polynomial functionsandgraphs
This document discusses polynomial functions and their graphs. It defines polynomial functions as functions of the form P(x) = anxn + an-1xn-1 + ... + a1x + a0, where an is the leading coefficient. The degree of the polynomial determines features of its graph like the maximum number of x-intercepts. The leading coefficient test determines the end behavior of the graph. Key features of polynomial graphs are intercepts, extrema, and end behavior.
Graph of functions
The document discusses different types of functions and their graphs. It provides examples of constant functions, linear functions, quadratic functions, and absolute value functions. It shows how to graph each type of function by plotting points and describes their domains and ranges. For linear functions, the domain is all real numbers and the range is also all real numbers. For quadratic functions, the graph is a parabola and the range only includes positive values. Absolute value functions have a domain of all real numbers and a range that is positive and excludes negative values below the constant.
Similar to Alg1 ch1101example123 (20)
1. Graph the exponential function by hand. Identify any asymptotes.docx
1. Graph the exponential function by hand. Identify any asymptotes.docx
Function evaluation, termination, vertical line test etc
Function evaluation, termination, vertical line test etc
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Alg1 ch1101example123
- 1. EXAMPLE 1 SOLUTION STEP 1 Graph the function y = 3 x and identify its domain and range. Compare the graph with the graph of y = x . Make a table. Because the square root of a negative number is undefined, x must be nonnegative. So, the domain is x ≥ 0 . STEP 2 Plot the points. Graph a function of the form y = a x
- 2. EXAMPLE 1 STEP 3 STEP 4 Draw a smooth curve through the points. From either the table or the graph, you can see the range of the function is y ≥ 0 . Graph a function of the form y = a x Compare the graph with the graph of y = x . The graph of y = 3 x is a vertical stretch (by a factor of 3 ) of the graph of y = x .
- 3. EXAMPLE 2 Graph a function of the form y = a x SOLUTION Graph the function y = –0.5 x and identify its domain and range. Compare the graph with the graph of y = x . To graph the function, make a table, plot the points, and draw a smooth curve through the points. The domain is x ≥ 0 . The range is y ≤ 0 .The graph of y = –0.5 x is a vertical shrink (by a factor of 0.5 ) with a reflection in the x- axis of the graph of y = x .
- 4. EXAMPLE 3 Graph a function of the form y = x + k SOLUTION Graph the function y = x + 2 and identify its domain and range. Compare the graph with the graph of y = x . To graph the function, make a table, then plot and connect the points. The domain is x ≥ 0 . The range is y ≥ 2 .The graph of y = x + 2 is a vertical translation (of 2 units up) of the graph of y = x .
- 5. GUIDED PRACTICE for Examples 1, 2 and 3 Graph the function and identify its domain and range. Compare the graph with the graph of y = x . 1. y = 2 x Domain: x ≥ 0 , Range: y ≥ 0 Vertical stretch by a factor of 2 ANSWER
- 6. GUIDED PRACTICE for Examples 1, 2 and 3 Graph the function and identify its domain and range. Compare the graph with the graph of y = x . 2. y = –2 x Domain: x ≥ 0 , Range: y ≤ 0 Vertical stretch by a factor of 2 and a reflection in the x -axis ANSWER
- 7. GUIDED PRACTICE for Examples 1, 2 and 3 Graph the function and identify its domain and range. Compare the graph with the graph of y = x . 3. y = x – 1 Domain: x ≥ 0 , Range: y ≥ 0 – 1 Vertical translation of 1 unit down ANSWER
- 8. GUIDED PRACTICE for Examples 1, 2 and 3 Graph the function and identify its domain and range. Compare the graph with the graph of y = x . 4. y = x + 3 Domain: x ≥ 0 , Range: y ≤ 0 Vertical translation of 3 units up ANSWER