Graphing Rational Functions
Graphing rational functions involves plotting the graph of a function that can be expressed as the ratio of two polynomials. Key steps include identifying vertical and horizontal asymptotes, finding x- and y-intercepts, and determining the behavior of the function as x approaches positive and negative infinity. Understanding the behavior of rational functions helps in visualizing their graphs and analyzing their properties.
6 Key excerpts on "Graphing Rational Functions"
- J. Rosebush, Stu Schwartz(Authors)
- 2016(Publication Date)
- Research & Education Association(Publisher)
...PART II FUNCTIONS, GRAPHS, AND LIMITS Chapter 2 Analysis of Graphs I. ANALYSIS OF GRAPHS A. Basic Functions—you need to know how to graph the following functions and any of their transformations by hand. 1. Polynomials, absolute value, square root functions 2. Trigonometric functions 3. Inverse trigonometric functions and their domain and range 4. Exponential and Natural Logarithmic functions 5. Rational functions 6. Piecewise functions 7. Circle Equations i. Upper semicircle with radius a and center at the origin:. This is a function. For example, ii. Lower semicircle with radius a and center at the origin:. This is a function. For example, iii. Circle with radius a and center at the origin: x 2 + y 2 = a 2. This is not a function since some x -values correspond to more than one y -value. For example, x 2 + y 2 = 9 iv. Circle with radius a and center at (b, c): (x – b) 2 + (y – c) 2 = a 2. This is not a function either. For example, (x – 2) 2 + (y + 3) 2 = 9 8. Summary of Basic Transformations of Functions A. Making changes to the equation of y = f (x) will result in changes in its graph. The following transformations occur most often. B. For trigonometric functions, f (x) = a sin(bx + c) + d or f (x) = a cos(bx + c) + d, a is the amplitude (half the height of the function), b is the frequency (the number of times that a full cycle occurs in a domain interval of 2 π units), is the horizontal shift and d is the vertical shift. Keep in Mind.... The reciprocal of sin(x),, is equivalent to csc(x), whereas sin –1 (x) is the inverse of sin(x), which is the reflection of sin(x) in the line y = x. When changing a function by adding a positive constant to x, the graph will shift to the left, not the right. The graph shifts to the right a units when a is subtracted from x. When graphing a function on the calculator (TI-83 or TI-84), make sure that all the plots are turned off; otherwise you risk getting an error and not being able to graph...
- Stu Schwartz(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...The discriminant b 2 − 4 ac will tell you how many solutions the quadratic has: EXAMPLE 11: Find the domain of SOLUTION:. The domain is all allowable values of x, which means (2 x − 3)(3 x + 2) ≠ 0. Then 2 x − 3 ≠ 0 and 3 x + 2 ≠ 0. So EXAMPLE 12: If y = 5x 2 − 3 x + k, for what values of k will the quadratic have two real solutions? SOLUTION: 9. Asymptotes Rational functions in the form of may have vertical asymptotes, lines that the graph of the curve approach, but never cross. To find the vertical asymptotes, factor out any common factors of the numerator and denominator, reduce if possible, and then set the denominator equal to zero and solve. Horizontal asymptotes are lines that the graph of the function approaches when x gets very large or very small. While you learn how to find these in calculus, a rule of thumb is that if the highest power of x is in the denominator, the horizontal asymptote is the line y = 0. If the highest power of x is both in the numerator and the denominator, the horizontal asymptote will be the line. If the highest power of x is in the numerator, there is no horizontal asymptote. EXAMPLE 13: Find any vertical and horizontal asymptotes for the graph of SOLUTION: Since the denominator factors into (x − 3)(x + 2), there are vertical asymptotes at x = 3 and x = −2. Since there the highest power of x is 2 in both the numerator and the denominator, there is a horizontal asymptote at. This is confirmed by the graph on the next page. Note that the curve actually crosses its horizontal asymptote on the left side of the graph. 10. Inverses No topic in math confuses students more than inverses. If a function is a rule that maps x to y, an inverse is a rule that brings y back to the original x. If a point (x, y) is a point on a function f, then the point (y, x) is on the inverse function f −1. If a function is given in equation form, to find the inverse, replace all occurrences of x with y and all occurrences of y with x. If possible, solve for y...
- Lawrence S. Leff, Christina Pawlowski-Polanish(Authors)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...7.17 Graph of EXAMPLE Since the degrees of the numerator and the denominator of are equal, is a horizontal asymptote of the graph, as shown in Figure 7.17. SLANT ASYMPTOTES If the degree of the numerator of a rational function is 1 more than the degree of the denominator, its graph has a slant asymptote. The graph of has a slant asymptote. To find its equation, do the long division: 1. Divide the numerator by the denominator: The answer is 2. Delete the remainder from the answer: 3. Write the equation of the slant asymptote by setting y equal to the remaining part of the quotient: y = 3 x. Figure 7.18 shows the graph of the function with its asymptote. FIGURE 7.18 Graph of HOLES Graphs of rational functions are discontinuous as a result of asymptotes or removable discontinuities, also known as “holes.” A rational function has a hole if there is a common factor in the numerator and denominator that can be removed and then graphed using an open circle, or “hole.” EXAMPLE To find all discontinuities (asymptotes and holes) for the graph of, first notice that the degrees of the numerator and denominator are equal and so y = 1 is a horizontal asymptote. By setting the denominator equal to zero, it would appear that there are vertical asymptotes at x = 3 and x = –3. However, if you factor the numerator and denominator,. There is a common factor of (x – 3) that can be removed, but it doesn’t disappear. At x = 3, you can find the corresponding y -value using the remaining factors of f (x) and graph a hole to show that there is a removable discontinuity. Hole at. The graph would be as follows: EXERCISE 1 Find all of the discontinuities for. SOLUTION Factoring the numerator and denominator,. Removing the common factor, (x – 1), tells you that there is a hole at x = 1. Using the remaining factors, the y -coordinate is found by evaluating. There is a hole at (1, 1)...
eBook - ePub- Thomas Pfaff(Author)
- 2019(Publication Date)
- Chapman and Hall/CRC(Publisher)
...2 Functions and Their Graphs We define and use functions throughout this book, and in this chapter we focus on functions and their associated graphs. We provide examples of commonly used mathematical functions: piecewise, step, parametric, and polar. We broaden the use of function by using the geometric definition to obtain a parabola, discuss functions that return functions, and create a function that returns Pythagorean triples. In each case we graph the function; we keep our graphs basic leaving chapter 3 for further details on graphing, except for our last example where we create a checkerboard graph. The command for creating a function is function() {}, where the variable(s) are listed inside the parenthesis and the function is defined within the braces. In our first example, we define the function f to be x 2 sin (x) and evaluate it at 3 with f(3). There are a number of predefined functions such as abs, sqrt, the trigonometric functions, hyperbolic functions, log for the natural log, log10, log2, and the exponential function exp. So, for example, sin(x) is available to use in our definition of f. Note that * must be used for multiplication as we cannot simply juxtapose objects. R Code > f=function(x){x ^ 2*sin(x)} > f(3) [1] 1.27008 We can plot our function with curve. The first three arguments must be the function, the lower value for the independent variable, and the upper value for the dependent variable. The default range for the dependent variable is selected based on the minimum and maximum of the function on the given interval. There are numerous options, such as ylim for the y limits, lwd for the width (i.e., thickness) of the curve, col for the color of the curve, xlab and ylab for labeling the axis, and lty for the type of line (e.g., dashed, dotted). R Code > curve(f,-5,5) Our next two examples illustrate functions of two and three variables. The first returns the area of a rectangle given the length and width...
eBook - ePub- Mel Friedman(Author)
- 2012(Publication Date)
- Research & Education Association(Publisher)
...CHAPTER 5 Functions and Their Graphs CHAPTER 5 FUNCTIONS AND THEIR GRAPHS ELEMENTARY FUNCTIONS A function is any process that assigns a single value of y to each number of x. Because the value of x determines the value of y, y is called the dependent variable and x is called the independent variable. The set of all the values of x for which the function is defined is called the domain of the function. The set of corresponding values of y is called the range of the function. PROBLEM Is y 2 = x a function? SOLUTION Graph the equation. Note that x can have two values of y. Therefore, y 2 = x is not a function. PROBLEM Find the domain and range for y = 5 – x 2. SOLUTION First determine if there are any values that would make the function undefined (i.e., division by 0). There are none. Thus, the domain is the set of real numbers. The range can be found by substituting some corresponding values for x in the equation. The range is the set of real numbers less than or equal to 5. PROBLEM Evaluate f (1) for y = f (x) = 5 x + 2. SOLUTION f (x) = 5 x + 2 f (1) = 5(1) + 2 = 5 + 2 = 7 OPERATIONS ON FUNCTIONS Functions can be added, subtracted, multiplied, or divided to form new functions. a. (f + g) (x) = f (x) + g (x) b. (f – g) (x) = f (x) – g (x) c. (f × g) (x). = f (x) g (x) d. PROBLEM Let f (x) = 2 x 2 – 1 and g (x) = 5 x + 3. Determine the following functions: 1. f + g 2. f – g 3. f × g 4. SOLUTION COMPOSITE FUNCTION The composite function f ° g is defined (f ° g)(x) = f (g (x)). PROBLEM Given f (x) = 3 x and g (x) = 4 x + 2. Find (f ° g) (x) and (g ° f) (x). SOLUTION (f ° g) (x) = f (g (x)) =. 3(4 x + 2) = 12 x + 6 (g ° f) (x) = g (f (x)) = 4(3 x) + 2 = 12 x + 2 Note that (f ° g) (x) (g ° f) (x). PROBLEM Find (f ° g) (2) if f (x) = x 2 – 3 and g (x) = 3 x + 1. SOLUTION Substitute the value of x = 2 in g (x): Substitute the value of g (2) in f (x): f (7) = (7) 2 – 3 = 49 – 3 =...
- Caryl Lorandini(Author)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...12 FUNCTIONS AND GRAPHING WHAT YOU WILL LEARN • How to produce a table, an equation, and a graph of a linear function; • How to identify the slope of a line as positive or negative from a graph; • How to determine the slope of a linear function, given an equation not necessarily in slope-intercept form; • How to interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; • How to compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions); • A function is a rule that assigns to each input exactly one output; • About domain and range, and which is allowed to have numbers repeat; • How to use the vertical line test to determine a function; • How to determine if functions are linear or non-linear; • How to identify linear functions from their equation, table, and graph. SECTIONS IN THIS CHAPTER • What Is Coordinate Geometry? • How Do We Graph a Line from a Table of Values? • What Is Slope? • How Do We Graph a Line Using Slope? • What Is a. Function? • What Is the Rule of Four? • How Can We Tell If a Function Is Linear or Non-linear? DEFINITIONS Axis A horizontal or vertical line used in the Cartesian coordinate system to locate a point on the coordinate graph. Axes The horizontal and vertical lines dividing a coordinate plane into four quadrants. Cartesian (coordinate) plane The plane formed by a horizontal axis and a vertical axis, often labeled the x -axis and y -axis, respectively...
