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Graphs and Functions

Parabolas

1. State the key features (vertex, focus, directrix, direction of opening, and axis of symmetry) of each parabola, and sketch the graph. a) ysquared-2x+4y-7=0 2. Find the equation of each parabola. a) parabola with focus (-2,-1) and directrix y=5 b) ysquared=-6x translated according to ((x,y)arrow(x-2, y+4)

Find the interval on which the function

See attached file for full problem description. Find the interval on which the function 2 2 ( ) ( 2) ( 3) f x x x = − + is increasing and decreasing. Sketch the graph of y = f(x), and identify any local maxima and minima. Any global extrema should also be identified.

Problem set

Question 1 Consider the functions f(x) = x^2 and g(x) = square root of x, both with domain and co-domain R+, the set of positive real numbers. Are f and g inverse functions? Give a brief reason. Question 2 Given the Hamming distance function f: A X A -> Z defined on pairs of 8-bit strings, (where A is the set o

Measurable Functions

Show that a function f is measurable IF AND ONLY IF there exists a sequence (f_m) of set functions such that f(x)=lim f_m(x) for almost all x. Please make sure to show the proof in both directions.

Graphing, Domain and Range

Given the table below, graph the function, identify the graph of the function (line, parabola, hyperbola, or exponential), explain your choice, and give the domain and range as shown in the graph, and also the domain and range of the entire function. x -2 -1 0 1 2 y .111 .333 1 3 9 See attachment

Value of function at a given point

1. The number of 4-year college, public and private, in the period 1980-1996 can be modeled by f(x)=0.0003x^3 - 0.007x^2+0.058x+1.957 0 less than or equal to X less then or equal to 16 Where X is the number of years since 1980 and f(x) is the number of 4-year colleges measured in thousands. Determine the average number

graph the functions with data points

For the function, y = ___1____ x - 2 a) Give the y values for x = -2, -1, 0, 1, 2, 3. Answer: Show work in this space. b) Using these points, draw a curve. Show graph here.

Comparing Relations and Functions; AIDS, Heart Disease, Trends and Forecasting

1. In the real world, what might be a situation where it is preferable for the data to form a relation but not a function? 2. There is a formula that converts temperature in degrees Celsius to temperature in degrees Fahrenheit. You are given the following data points: Fahrenheit Celsius Freezing point of water 32 0 Bo

Graphs and Solving Linear Equations Word Problems (20 Problems)

Solve the following equations for the unknown. 1. 5x = 20 2. 7x - 3 = 18 Graph the following equations; calculate the slope, x-intercept, and y-intercept, and label the intercepts on the graph. 3. y = x + 3 4. y = -2x - 7 5. 2x + 3y = 9 6. A consumer electronics c

Break even analysis: Labor intensive firm and capital intensive firms

Draw two break-even graphs-one for a conservative firm using labor-intensive production and another for a capital-intensive firm. Assuming these companies compete within the same industry and have identical sales, explain the impact of changes in sales volume on both firms' profits. Although no example is provided in the

Comparing Sequences and Functions

Using the index of a sequence as the domain and the value of the sequence as the range, is a sequence a function? Include the following in your answer: 1. Which one of the basic functions (linear, quadratic, rational, or exponential) is related to the arithmetic sequence? 2. Which one of the basic functions (linear, quadrat

Equations of Straight Lines

1. Graph line with equation y=-4x-2. 2. 2x=8y=17=0 3. Graph the line with slope -2 passing through the point (-3,4). 4. Find slope of the line graphed (-44, 28) (9, -12). 5. x=9 graph the line 6. 6x-9y+7=0. 7. 2x=5y+3 8. Write an equation of line (0, -4). 9. A line passes through the point (6, -6

Asymptotes, Descartes's Rule and coordinates of the vertex

1. f(x) = x^3 + x^2 - 4x - 4. (a) What is the end behavior of this function? (Does it go up or down to the left? Does it go up or down the right?) (b) What is the maximum number of turning points for this function? 2. Give the coordinates of the vertex of the parabola y = (x + 2)^2 + 5. 3. Use Descartes's Rule of Sig

Simplifying and Solving Equations with Exponents

1. Simplify the expression using properties of exponents. Answer should contain positive exponents only. (3x^2y^3)^2(3xy)^2 / (2x^4y^3)^-3 2. Solve for X: 3/x - 1/x+2 = -2/3x+6 3. Use the quadratic formula to solve for x: x^2 -4x + 2 = 0 4. Solve for x by factoring: x^3 + x^2 - 6x =

Euler Function

(See attached file for full problem description) Conjecture: Suppose that m and n are positive integers. If gcd(m,n)=1, then . a. If m and n are positive integers and k is any integer, show that gcd(k,mn)=1 if and only if gcd(k,m)=1 and gcd(k,n)=1. b. Suppose gcd(m,n)=1. Prove that establishes a bijection between

Exponential Growth and Decay Projects

Think of one real situation that involves exponential growth and that involves exponential decay. for each example, your project should include the following: * Paragraph - Briefly explain the situation. You may make up your own information, but make it realistic. Include the facts needed to write an equation. * Equation -

Real-Life Applications of Hyperbolas and Parabolas

One of the civil engineers you interviewed for your article works for a company which specializes in bridge construction projects. In the process of designing suspension bridges, they must account for many variables in the modeling. Some of these variables include the bridge span; the force of the typical water currents wearing

Discriminants

When using the quadratic formula to solve a quadratic equation (ax2 + bx + c = 0), the discriminant is b2 - 4ac. This discriminant can be positive, zero, or negative. Create three unique equations where the discriminant is positive, zero, or negative. For each case, explain what this value means to the graph of y = ax2 + bx +

Radicals

(See attached file for full problem description with equations) --- 1) Solve the following equations. a) Answer: Show work in this space. b) . Answer: Show work in this space. c) . Answer: Show work in this space. 2) Is an identity (true for all values of x)? Answer:

One-to-One and Onto Functions

Suppose that X and Y are finite sets, with m and n elements respectively. Suppose further that the function f : X → Y is one-to-one and the function g : X →Y is onto. (i) Use the function f to show that m ≤ n. (ii) Use the function g to show that m ≥ n. (iii) Is f : X → Y onto? Justify your assertion. (iv) Is th

Input-output Tables and Equations of Lines, Slope and Intercept

Decide whether the relation is a function. 1- (6,5), (4,3), (-2,3), (0,-1) , (-1,2), (-4,-5), (-3,4) 2- (4,4), (3,3), (-1,1), (6,6), (1,-2) make an input-output table for the function rule. Use a domain of -10,-5,0,5, and 10. Identify the range. y=8x+1 y=-6x y=x square 2+5 write a function that relates and x and