This week in precalc 12 we learned about adding and subtracting rational fractions.
To add or subtract rational expressions with unlike denominators, first find the LCM of the denominator. The LCM of the denominators of fraction or rational expressions is also called least common denominator , or LCD. Write each expression using the LCD. Make sure each term has the LCD as its denominator. Add or subtract the numerators. Simplify as needed.
In order to find the vertical asymptotes of a rational function, you need to have the function in factored form. You also will need to find the zeros of the function. For example, the factored function y=x+2(x+3)(x−4) has zeros at x = – 2, x = – 3 and x = 4.
If the numerator and denominator have no common zeros, then the graph has a vertical asymptote at each zero of the denominator. In the example above y=x+2(x+3)(x−4), the numerator and denominator do not have common zeros so the graph has vertical asymptotes at x = – 3 and x = 4.
If the numerator and denominator have a common zero, then there is a hole in the graph or a vertical asymptote at that common zero.
Examples:
1. y=(x+2)(x−4)x+2 is the same graph as y = x – 4, except it has a hole at x = – 2.
2.y=(x+2)(x−4)(x+2)(x+2)(x−4) is the same as the graph of y=1x+2,except it has a hole at x = 4. The vertical asymptote is x = – 2.
To Find Horizontal Asymptotes:
Graphing absolute value linear and quadratic equations:
1. Identify the vertex (not necessary for Linear functions). • Absolute Value functions: This is the point at which the definition of the function changes. Solve for the interval on which the expression inside the absolute value is greater than or equal to zero. • Quadratic functions: This can be determined from the standard form, y = b(x − h) 2 + k, and is the point (h, k). A Quadratic function can be written in standard form by completing the square. • Square Root functions: This can be determined from one of the two standard forms, y = b √ x − h + k or y = b √ h − x + k. Each Square Root function can only be written in one of the two standard forms. In either case, the vertex is the point (h, k).
2. Find the x-intercept(s), if any. For each type of function, this means solving the equation f(x) = 0. • Linear functions: There will always be exactly one x-intercept. • Absolute Value and Quadratic functions: There may be zero, one or two x-intercepts. • Square Root functions: There may be zero or one x-intercepts.
3. Find the y-intercept, if it exists. For each type of function, this means evaluating f(0). • Linear, Absolute Value and Quadratic functions: There will always be one y-intercept. • Square Root functions: There may be zero or one y-intercept.
4. Determine as many other points as needed (if any) to accurately sketch the graph. • Linear functions: We need two points in total. • Absolute Value functions: We need one point on each side of the vertex. • Quadratic functions: We need two points on each side of the vertex. • Square Root functions: We need two points other than the vertex.
5. Plot and label all of the points, and sketch the graph.