Roots With Calc
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This document provides instructions for using a graphing calculator to find the roots, vertex, axis of symmetry, and turning point of quadratic functions. It gives examples of quadratic functions and has students use the calculator to find the nature of the roots, the roots themselves, the y-intercept, and the maximum or minimum point. Students are then asked to complete homework problems listing these values for additional quadratic functions and sketching the graphs.
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The document discusses using a graphing calculator to find the roots and vertex of quadratic functions. It provides examples of quadratic functions and steps to determine the nature of the roots, find the actual roots, y-intercept, axis of symmetry, turning point, and whether the graph is concave up or down in order to sketch the graph. The goal is to demonstrate how to use a graphing calculator to analyze quadratic functions.
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The document provides instructions for using a graphing calculator to find the roots, vertex, maximum or minimum value of quadratic functions. It lists steps like determining the nature of the roots, finding the roots, y-intercept, axis of symmetry, and turning point. Examples of quadratic functions are given, along with homework problems that ask students to provide details about the roots, intercepts, vertex and concavity for each function.
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The document provides instructions for using a graphing calculator to find the roots, vertex, maximum or minimum value of quadratic functions. It lists steps like determining the nature of the roots, finding the roots, y-intercept, axis of symmetry, and turning point. Examples of quadratic functions are given, along with homework problems that ask students to provide details about the roots, intercepts, vertex and concavity for each function.
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The document contains several mathematics problems involving:
1) Calculating values of expressions with directed numbers, integers, fractions, decimals, square roots, cubes, and algebraic expressions.
2) Solving equations and inequalities involving one or more variables.
3) Working with indices, statistics, functions, and graphs. Problems cover topics such as mean, median, mode, frequency tables, pie charts, and plotting points to graph functions.
Class XII CBSE Mathematics Sample question paper with solution
Class XII CBSE Mathematics Sample question paper with solutionPratima Nayak ,Kendriya Vidyalaya Sangathan
The document is a sample question paper for Class XII Mathematics. It consists of 3 sections - Section A has 10 one-mark questions, Section B has 12 four-mark questions, and Section C has 7 six-mark questions. All questions are compulsory. The paper tests concepts related to matrices, trigonometry, calculus, differential equations, and vectors. Internal choices are provided in some questions. Calculators are not permitted.Math iecep
The document contains a collection of math and science problems related to topics like calculus, vectors, probability, statistics, and more. It includes over 50 multiple choice questions testing concepts like Laplace transforms, vector operations, differential equations, probability distributions, and calculating averages, means, and standard deviations from data sets. The questions are in Filipino with translations provided and cover a wide range of computational, algebraic, and conceptual problems across several domains of math, science, and engineering.
1) Use properties of logarithms to expand the following logarithm.docx
1) Use properties of logarithms to expand the following logarithmic expression as much as possible.
Logb (√xy3 / z3)
A. 1/2 logb x - 6 logb y + 3 logb z
B. 1/2 logb x - 9 logb y - 3 logb z
C. 1/2 logb x + 3 logb y + 6 logb z
D. 1/2 logb x + 3 logb y - 3 logb z
2) Solve the following logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, to two decimal places, for the solution.
2 log x = log 25
A. {12}
B. {5}
C. {-3}
D. {25}
3) Write the following equation in its equivalent logarithmic form.
2-4 = 1/16
A. Log4 1/16 = 64
B. Log2 1/24 = -4
C. Log2 1/16 = -4
D. Log4 1/16 = 54
4) Use properties of logarithms to condense the following logarithmic expression. Write the expression as a single logarithm whose coefficient is 1.
log2 96 – log2 3
A. 5
B. 7
C. 12
D. 4
5) Use the exponential growth model, A = A0ekt, to show that the time it takes a population to double (to grow from A0 to 2A0 ) is given by t = ln 2/k.
A. A0 = A0ekt; ln = ekt; ln 2 = ln ekt; ln 2 = kt; ln 2/k = t
B. 2A0 = A0e; 2= ekt; ln = ln ekt; ln 2 = kt; ln 2/k = t
C. 2A0 = A0ekt; 2= ekt; ln 2 = ln ekt; ln 2 = kt; ln 2/k = t
D. 2A0 = A0ekt; 2 = ekt; ln 1 = ln ekt; ln 2 = kt; ln 2/k = toe
6) Find the domain of following logarithmic function.
f(x) = log (2 - x)
A. (∞, 4)
B. (∞, -12)
C. (-∞, 2)
D. (-∞, -3)
7) An artifact originally had 16 grams of carbon-14 present. The decay model A = 16e -0.000121t describes the amount of carbon-14 present after t years. How many grams of carbon-14 will be present in 5715 years?
A. Approximately 7 grams
B. Approximately 8 grams
C. Approximately 23 grams
D. Approximately 4 grams
8) Use properties of logarithms to expand the following logarithmic expression as much as possible.
logb (x2 y) / z2
A. 2 logb x + logb y - 2 logb z
B. 4 logb x - logb y - 2 logb z
C. 2 logb x + 2 logb y + 2 logb z
D. logb x - logb y + 2 logb z
9) The exponential function f with base b is defined by f(x) = __________, b > 0 and b ≠ 1. Using interval notation, the domain of this function is __________ and the range is __________.
A. bx; (∞, -∞); (1, ∞)
B. bx; (-∞, -∞); (2, ∞)
C. bx; (-∞, ∞); (0, ∞)
D. bx; (-∞, -∞); (-1, ∞)
10) Approximate the following using a calculator; round your answer to three decimal places.
3√5
A. .765
B. 14297
C. 11.494
D. 11.665
11) Write the following equation in its equivalent exponential form.
4 = log2 16
A. 2 log4 = 16
B. 22 = 4
C. 44 = 256
D. 24 = 16
12) Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents.
31-x = 1/27
A. {2}
B. {-7}
C. {4}
D. {3}
13) Use properties of logarithms to expand the following logarithmic expression as much as possible.
logb (x2y)
A. 2 logy x + logx y
B. 2 logb x + logb y
C. logx - logb y
D. logb x – ...
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Roots With Calc
- 1. Aim: How do we find the roots and vertex using the graphing calculator? MB43 3/10/09 Lomas Do Now: Find the roots, y-int, axis of symmetry, turning point, max or min, sketch graph: y = x 2 - x - 6 HW Review: 7, 19, 22 Determine the nature of the roots of y = 2x 2 - 5x - 6 HW: For each list the: a) Nature of Roots b) Roots c) y-intercept d) Axis of Symmetry e) Turning Piont f) Concave up or down g) Sketch the graph 1) y = x 2 + 8x + 6 2) f(x) = 2x 2 + 3x - 2
- 2. Aim: How do we find the roots and vertex using calculator the graphing? MB43 3/10/09 Lomas Use the calculator to find roots, max or min, and vertex y = 2x 2 - 5x - 6 HW: For each list the: a) Nature of Roots b) Roots c) y-intercept d) Axis of Symmetry e) Turning Piont f) Concave up or down g) Sketch the graph 1) y = x 2 + 8x + 6 2) f(x) = 2x 2 + 3x - 2
- 3. Aim: How do we find the roots and vertex using calculator the graphing? MB43 3/10/09 Lomas y = 2x 2 + 3x - 4 y = x 2 + 3x - 10 y = x 2 + 8x + 17 1) 3) 2) Use a calculator to find a)roots, b)max or min, and c)turning point HW: For each list the: a) Nature of Roots b) Roots c) y-intercept d) Axis of Symmetry e) Turning Piont f) Concave up or down g) Sketch the graph 1) y = x 2 + 8x + 6 2) f(x) = 2x 2 + 3x - 2
- 4. Aim: How do we find the roots and vertex using calculator the graphing? MB43 3/10/09 Lomas Determine the nature of the roots, then find them: y = x 2 + 3x - 10 0 = x 2 + 8x + 17 1) 3) 2) 4 = 2x 2 + 3x HW: For each list the: a) Nature of Roots b) Roots c) y-intercept d) Axis of Symmetry e) Turning Piont f) Concave up or down g) Sketch the graph 1) y = x 2 + 8x + 6 2) f(x) = 2x 2 + 3x - 2