Question

Multiple Choice Identify the \(y\) -intercept of the graph of $$ R(x)=\frac{6(x-1)}{(x+1)(x+2)} $$ (a) -3 (b) -2 (c) -1 (d) 1

Short answer

The y-intercept is -3.

Step-by-step solution

- Understand the y-intercept

The y-intercept of a function is the point where the graph of the function crosses the y-axis. This occurs when the value of x is 0.

- Substitute x=0 into the function

To find the y-intercept, substitute x=0 into the function R(x). The function is: \[ R(x)=\frac{6(x-1)}{(x+1)(x+2)} \] Replace x with 0: \[ R(0)=\frac{6(0-1)}{(0+1)(0+2)} \]

- Simplify the expression

Now simplify the expression: \[ R(0)=\frac{6(-1)}{(1)(2)}=\frac{-6}{2}=-3 \]

- Identify the answer

The y-intercept is -3. Look at the multiple-choice options to select the correct answer.

Key concepts

Finding Intercepts

The intercept points are crucial for understanding the behavior of functions. Intercepts are the points where the graph of a function crosses the axes. There are two types: the x-intercept and the y-intercept. The y-intercept is particularly important because it tells us the value of the function when x = 0. To find the y-intercept of any function, you simply substitute x = 0 into the function and solve for y. For the function in the exercise, \[ R(x)=\frac{6(x-1)}{(x+1)(x+2)}. \] we set x = 0: \[ R(0)=\frac{6(0-1)}{(0+1)(0+2)}. \] Simplifying, we get \[ R(0)=\frac{-6}{2}=-3. \] This tells us that the y-intercept is -3.

Rational Functions

Rational functions are fractions of polynomials. They come in the form \[ R(x) = \frac{P(x)}{Q(x)} \] where P(x) and Q(x) are polynomials. These functions can behave in interesting ways near their intercepts and asymptotes. For example, if Q(x) = 0, then the function has a vertical asymptote, which means that the function's value can go to infinity. One important skill in analyzing rational functions is finding the intercepts. This helps in getting a basic sketch of the graph. Always remember that for the y-intercept, x is set to zero: \[ R(0)=\frac{P(0)}{Q(0)}. \] Rational functions often have restrictions, usually set by the denominator, as it cannot be zero.

Graph Analysis

Graph analysis involves understanding the main features of a graph, such as intercepts, asymptotes, increase or decrease, and other behaviors. Begin by finding the intercepts, as they provide clear points where the graph crosses the axes. In the case of rational functions, it’s also critical to look for vertical asymptotes where the denominator equals zero. These vertical lines indicate values that the function cannot take on. For our function, \[ R(x)=\frac{6(x-1)}{(x+1)(x+2)}, \] there might be asymptotes at x = -1 and x = -2. By combining these points with the intercepts, you can better predict the behavior and shape of the graph. Thus, understanding the intercepts gives you a valuable starting point for more detailed graph analysis.