MATA30H3 Lecture Notes - Algebraic Equation, Asymptote

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7 May 2013

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The reciprocal of a linear function has the form: f(x) = 1 / kx c. The restriction on a domain of a reciprocal linear function can be determined by finding the value of x that makes the denominator equal to zero, that is x = c / k. The vertical asymptote of a reciprocal linear function has an equation of the form x = k / c. The horizontal asymptote of a reciprocal linear function has equation y = 0. If k > 0, the left branch of a reciprocal linear function has a negative, decreasing slope, and the right branch has a negative, increasing slope. If k < 0, the left branch of a reciprocal linear function has a positive, increasing slow, and the right branch has a positive, decreasing slope. Rational functions can be analyzed using key features: asymptotes,intercepts, slope (positive or negative, increasing or decreasing),domain, range, and positive and negative intervals.

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must understand functions to be able linearize. First let's review what graphs of certain functions looks like. Sketch the shape of each type of y vs. x function below. k is listed as a generic constant of proportionally. Linear y = kx Inverse y = k/x Inverse Square y = k/x^2 Power y = kx^2 You will notice that only the linear function is a straight line. We can easily find the slope of our line by measuring the rise and dividing it by the run of the graph or calculating it using two points. The value of the slope should equal the constant, k, from the equation. Finding k is a bit more challenging in the last three graphs because the slope isn't constant. This should make sense since your graphs aren't linear. So how do you calculate the constant, k? You need to transform the non-linear graph into a linear graph in order to calculate a constant slope. You can accomplish this by transforming one or both of the axes for the graph. The hardest part is figuring out which axes to change and how to change them. The easiest way to accomplish this task is to solve your equation for the constant. Note in the examples from the last page there is only one constant, but this process could be done for other equations with multiple constants. Instead of solving for a single constant, put all of the constants on one side of the equation. When you solve for the constant, the other side of the equation should be in fraction form. This fraction gives the rise and run of the linear graph. Whatever is in the numerator is the vertical axis and the denominator is the horizontal axis. If the equation is not in fraction form, you will need to inverse one or more of the variables to make a fraction. First let's solve each equation to figure out what we should graph. Then look below at the example and complete the last one, a sample AP question, on your own. State what should be graphed in order to produce a linear graph to solve for k. Inverse Graph Vertical Axis: _____ Horizontal Axis: _____ Inverse Square Graph Vertical Axis: _____ Horizontal Axis: _____ Power (Square) Graph Vertical Axis: _____ Horizontal Axis: _____

Algebra

Show transcribed image text Use a transformation of the graph of y = x3 to graph the function. h(x)=4 - (x - 2)5 Choose the correct graph below. Find the vertical, horizontal, and oblique asymptotes, if any, for the given rational function. Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) is / are x = (Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression.) There is no vertical asymptote. Select the correct choice below and fill in any answer boxes within your choice. The horizontal asymptote(s) is / are y = (Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression.) There is no horizontal asymptote. Select the correct choice below and fill in any answer boxes within your choice. The oblique asymptote{s)is / are y = (Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression.) There is no oblique asymptote. Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f(x) = 6x2 + 12x - 3 The quadratic function has a value. The value is For the data given below, answer parts (a) through (d). Find the equation of the line containing the first and the last data points. (Type your answer in slope-intercept form. Use integers or fractions for any numbers in the expression.) Draw a scatter diagram and the line found in part (a) on the same axes. Choose the correct graph below. Use a graphing utility to find the line of best fit. Which of the following is the equation of the line of best fit? Use a graphing utility to draw the scatter diagram and graph the line of best fit on it. Choose the correct graph below. Complete the square of the given quadratic expression. Then, graph the function using the technique of shifting. f(x) = x2 - 8x + 15 Complete the square by entering the correct numbers into the expression below. Choose the correct graph below. Analyze the graph of the function. What is the domain of R(x)? What is the equation of the vertical asymptote(s) of R(x)? Select the correct choice below and fill in any answer boxes within your choice. What is the equation of the horizontal or oblique asymptote of R(x)? Select the correct choice below and fill in any answer boxes within your choice. Choose the correct graph for R(x) below. Find k such that f(x) = x3 - kx2 + kx + 4 has the factor x - 2. k = (Simplify.) F(x) = 4x4 - 8X2 + 3 Determine whether F is even. odd. or neither. There is a local minimum of - 1 at 1. Determine a second local minimum, (a) Determine whether F is even. odd. or neither. Neither Even Odd There is a local minimum of - 1 at 1. Determine a second local minimum. The second local minimum is . F(x) = 4x4 - 8X2 + 3 Determine whether F is even. odd. or neither. There is a local minimum of - 1 at 1. Determine a second local minimum, (a) Determine whether F is even. odd. or neither. Neither Even Odd There is a local minimum of - 1 at 1. Determine a second local minimum. The second local minimum is .The function f is defined as follows. Find the domain of the function. Locate any intercepts. Graph the function. Based on the graph, find the range. Is f continuous on its domain? The domain of the function f is (Type your answer in interval notation.) Locate any intercepts. Select the correct choice below and. if necessary, fill in the answer box to complete your choice. The intercept(s) is/are (Type an ordered pair. Use a comma to separate answers as needed.) There are no intercepts. Choose the correct graph of f(x) below. Is f continuous on its domain? Select the correct choice below and. if necessary, fill in the answer box to complete your choice. Find the difference quotient of f; that is. find for the following function. Be sure to simplify. Determine whether the given function is linear or nonlinear. If it is linear, determine the slope. Is the function a linear function? Determine the slope. Select the correct choice below and fill in any answer boxes within your choice. The slope of the function is .The function is not linear. Analyze the graph of the function. What is the domain of f(x)? What is the equation of the vertical asymptote(s) of f(x)? Select the correct choice below and fill in any answer boxes within your choice. x = (Use a comma to separate answers as needed. Type an integer or a fraction.) There is no vertical asymptote. What is the equation of the horizontal or oblique asymptote of f(x)? Select the correct choice below and fill in any answer boxes within your choice. y = (Simplify your answer.) There is no horizontal or oblique asymptote. Choose the correct graph for f(x) below. Use a transformation of the graph of y = x3 to graph the function. f(x) = (x - 6)5 + 5 Select the graph of f(x) = (x - 6)3 + 5. Determine whether the graph below is that of a function by using the vertical-line test. If it is, use the graph to find its domain and range. the intercepts, if any. any symmetry with respect to the x-axis: y-axis: or the origin. Is the graph that of a function? If the graph is that of a function, what are the domain and range of the function? Select the correct choice below and fill in any answer boxes within your choice. What are the intercepts? Select the correct choice below and fill in any answer boxes within your choice. (Type an ordered pair. Use a comma to separate answers as needed.) There are no intercepts. The graph is not a function. Determine if the graph is symmetrical. It is symmetrical with respect to the origin. It is symmetrical with respect to the x-axis. It is symmetrical with respect to the y-axis. The graph is not symmetrical. The graph is not a function. Solve the following inequality. Choose the correct solution.

turquoisewolf363

1. Give the domain of the function g(y) = 9/Y^(2+6Y+9) Select the correct choice below, and if necessary, fill in the answer box to complete your choice.

A.The domain of g is ( , ].

B.The domain of g is ( , ).

C.The domain of g is [ , ).

D.The domain of g is (-â,â)minusâ{ }.

2. Use a graphing calculator to find (a) the maximum value of f(x) and (b) the minimum value of f(x) for the indicated values of x. F(x)=-0.4x^3 -0.7x^2+5x-5 where -6â¤xâ¤4

The maximum value of f(x) in the intervals occurs when

The minimum value for X= Y=

3. Find the general linear equation (Ax+By+C= 0) of a straight line that passes through (negative 5â 55 ) and has a slope of â1\2 , and sketch the line.

The general linear equation is 0=

MATA30H3 Lecture Notes - Algebraic Equation, Asymptote

MATA30H3 Full Course Notes

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