Question

In Exercises 11–18, graph the function. State the domain and range. $$ y=\frac{10}{x+7}-5 $$

Short answer

The graph of the function \(y=\frac{10}{x+7}-5\) is a hyperbola that has been shifted 7 units to the left, stretched vertically by a factor of 10, and shifted down by 5 units. The domain of the function is all real numbers except -7 and the range is all real numbers except -5.

Step-by-step solution

Identify the Function Type

The given function \(y=\frac{10}{x+7}-5\) is a rational function, because it is a ratio of two polynomials, where the denominator isn't zero.

Graph the Function

To graph this function, it's helpful to know that it's a transformation of the function \(y=\frac{1}{x}\), which is a hyperbola. The +7 in the denominator shifts the graph 7 units to the left, the factor of 10 stretches the graph vertically by a factor of 10, and the -5 shifts the graph down by 5 units. The vertical asymptote of the graph is \(x=-7\) and the horizontal asymptote is \(y=-5\).

Determine the Domain and Range

The domain of the function is all real numbers except for the vertical asymptote. So the domain is \(x \neq -7\). The range of the function is all real numbers except for the horizontal asymptote, so the range is \(y \neq -5\).

Key concepts

Rational Functions

A rational function is a mathematical expression that represents the division of two polynomials. It's in the form of \( y = \frac{P(x)}{Q(x)} \) where both \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) \) is not equal to zero. These functions are interesting to graph, because they often exhibit significant features like asymptotes and intercepts.

Graphing a rational function starts by looking for these characteristics. For instance, the behavior of the graph around the values that make \( Q(x) = 0 \) is crucial because the function is undefined at these points. These values, where the graph tends to go off towards infinity, are closely tied to a concept called vertical asymptotes. Another interesting feature to investigate is the x-intercepts, which occur when \( P(x) = 0 \) and the y-intercepts which are found by evaluating the function when \( x = 0 \) if that's within the domain.

Domain and Range

Exploring the domain and range of a function involves identifying the set of possible inputs (the domain) and the set of possible outputs (the range). For rational functions, the domain is all real numbers except those that would make the denominator zero, because division by zero is undefined. In our example, this means \( x eq -7 \) since that would make the denominator zero and thus is excluded from the domain.

The range of a rational function can be trickier to determine but generally includes all real numbers that the function can output. This is all numbers the function can reach, except potentially for horizontal asymptotes which the function approaches but never touches. For the given function, \( y eq -5 \) establishes the range, indicating that the function will never output the value of -5, which corresponds to the horizontal asymptote of the function.

Asymptotes

Asymptotes are lines that the graph of a function approaches but never actually touches. These can be vertical, horizontal, or even oblique. With rational functions, vertical asymptotes occur at values of x that make the denominator zero (since the function cannot have a zero in the denominator). Horizontal asymptotes are found by examining the behavior of the function as x approaches infinity or negative infinity.

In our example, we have a vertical asymptote at \( x = -7 \) and a horizontal asymptote at \( y = -5 \). The function will get infinitely close to these lines but will never cross them. Identifying asymptotes is crucial for understanding the long-term behavior of the function and helps significantly in sketching its graph.

Function Transformation

Function transformations occur when we apply certain operations to an entire function, which results in shifting, stretching, compressing, or reflecting its graph. These operations can include adding or subtracting a number to the function's formula, which leads to vertical or horizontal shifts, or multiplying the function by a number, which results in vertical or horizontal stretches or compressions.

For instance, the given function \( y=\frac{10}{x+7}-5 \) is derived from the parent function \( y=\frac{1}{x} \) by shifting it 7 units to the left \( (x+7) \) and 5 units down \( (-5) \), plus vertically stretching it by a factor of 10 due to the multiplication by 10. By understanding these transformations, we can predict and graph the behavior of the function without plotting numerous points. This forms an essential tool for students as it simplifies the graphing process.