Graphing Rational Functions Quiz

The y-intercept of a function is the point where the graph of the function intersects the y-axis. To find the y-intercept, we substitute x=0 into the equation and solve for y. In this case, when x=0, the equation becomes y=(0+5)/(0-6) = 5/-6 = -5/6. Therefore, the y-intercept of y=(x+5)/(x-6) is (0, -5/6).

An asymptote is an imaginary line that a function never touches. It is a line that the graph of a function approaches but does not intersect or cross. Asymptotes can be vertical, horizontal, or oblique, and they help to describe the behavior of a function as it approaches infinity or negative infinity. They provide information about the limits of a function and help to understand its overall shape and characteristics.

The vertical asymptote of a rational function occurs at the values of x that make the denominator equal to zero. In this case, the denominator is x-6, so the vertical asymptote occurs when x-6=0. Solving for x, we find that x=6. Therefore, the vertical asymptote of the given function is x=6.

The horizontal asymptote of a function represents the value that the function approaches as x approaches positive or negative infinity. In this case, as x becomes very large or very small, the term (4x+1)/(x+2) approaches 4. Therefore, the horizontal asymptote of the function f(x) = (4x+1)/(x+2) is y = 4.

The domain of a function is the set of all possible values that the independent variable (x) can take. In this case, the function is y=1/(x+3). The denominator of the fraction cannot be zero, so x+3 cannot be equal to zero. Therefore, the value of x cannot be -3. Hence, the domain of the function is all real numbers except -3.

The horizontal asymptote of a function represents the value that the function approaches as x approaches positive or negative infinity. In this case, as x becomes very large or very small, the term -2x in the numerator will dominate the term x+1 in the denominator. Therefore, the function will approach -2 as x approaches infinity or negative infinity, making y = -2 the horizontal asymptote.

The given function is a rational function. In a rational function, the denominator cannot be equal to zero. Therefore, the domain of the function is all real numbers except for the values of x that make the denominator zero. In this case, the denominator is 4x^2-4, which can be factored as 4(x^2-1). Setting this equal to zero, we get x^2-1=0, which factors as (x-1)(x+1)=0. So, the values of x that make the denominator zero are x=-1 and x=1. Therefore, the domain of the function is all real numbers except x=-1 and x=1.

The asymptotes determined by looking at the denominator are the vertical asymptotes. This is because the vertical asymptotes occur at the values of x where the denominator is equal to zero. When the denominator is zero, the fraction becomes undefined, resulting in a vertical line that the graph approaches but never touches. Therefore, the correct answer is "Vertical."

The x-intercept of a function represents the point where the function intersects the x-axis. In this case, to find the x-intercept of the equation y=4x/(x+5), we set y=0 and solve for x. By substituting y=0 into the equation, we get 0=4x/(x+5). Multiplying both sides by (x+5), we eliminate the denominator and obtain 0=4x. Dividing both sides by 4, we find x=0. Therefore, the x=0 is the x-intercept of the given equation.

As x approaches infinity, the denominator x^2 becomes extremely large compared to the numerator 2x. Therefore, the fraction 2x/1-x^2 approaches 0. This can be seen by considering the limit of the function as x approaches infinity, where the numerator remains finite while the denominator grows without bound.