Graphing Rational Functions And Reciprocal Functions Quiz

Explanation
The given function does not have a horizontal shift. A horizontal shift refers to a transformation that moves the graph of a function horizontally left or right. Since the function does not have this type of shift, it does not apply to the given function.

Explanation
A vertical asymptote represents a value at which a rational function is undefined, meaning that the function cannot be evaluated at that particular value. This value is not included in the domain of the function. Therefore, the statement "A vertical asymptote shows a value at which a rational function is undefined. Thus, the value is not in the domain of the function." is true.

Explanation
A reciprocal function is defined as a function where the output is the reciprocal of the input. The reciprocal of a number is obtained by dividing 1 by that number. In a reciprocal function, the denominator of the fraction will always be the input value. Since division by zero is undefined, the denominator cannot be equal to zero. Therefore, a reciprocal function will never have values in its domain that result in the denominator being equal to zero. Hence, the statement is true.

Explanation
The x-intercept of a function is the value of x at which the function intersects the x-axis. To find the x-intercept, we set y equal to zero and solve for x. In this case, setting y equal to zero in the equation y = 4x/(x+5), we get 0 = 4x/(x+5). By cross-multiplying, we get 0 = 4x, which means x must be equal to 0. Therefore, x = 0 is the x-intercept of the equation y = 4x/(x+5).

Explanation
The given equation is in the form y = 1/x+2 - 3. This equation represents a rational function. The graph of this equation is a hyperbola that is shifted 2 units to the left and 3 units down from the standard reciprocal function y = 1/x. The -3 at the end of the equation represents the vertical shift downward by 3 units. Therefore, the correct answer is y = 1/x+2 - 3.

Explanation
A reciprocal function is not known as a slope function. A reciprocal function is a function that maps a non-zero number to its reciprocal, which is one divided by the number. On the other hand, a slope function refers to the rate of change of a function with respect to its input variable. These two concepts are distinct and not interchangeable. Therefore, the statement is False.

Explanation
An asymptote is not an imaginary line that a function always touches. In fact, an asymptote is a line that a function approaches but never quite reaches. It can be horizontal, vertical, or oblique, and it represents a limit or boundary for the function. Therefore, the correct answer is False.

Explanation
The horizontal asymptote of a function represents the value that the function approaches as x approaches positive or negative infinity. In this case, the function f(x) = -2x/(x+1) has a horizontal asymptote at y = -2. This means that as x gets larger and larger or smaller and smaller, the function approaches a value of -2.

Explanation
The vertical asymptote of a rational function occurs at values of x where the denominator is equal to zero. In this case, the denominator is x-6. Setting x-6 equal to zero and solving for x gives x=6. Therefore, the vertical asymptote of the function y=(x+5)/(x-6) is x=6.