Problem 57
Question
Sketch a graph of the rational function and find all intercepts and slant asymptotes. Indicate all asymptotes onthe graph. $$h(x)=\frac{4-x^{2}}{x}$$
Step-by-Step Solution
Verified Answer
The graph of the function \( h(x)=-x+4/x \) has x-intercepts at -2 and 2, a vertical asymptote at x=0, a slant asymptote at \( y=-x \) and no y-intercept.
1Step 1: Simplify the Rational Function
Simplify the rational function \( h(x)=\frac{4-x^{2}}{x} \) by dividing each term in the numerator by the denominator, getting two functions: \( h(x)=-x+4/x \)
2Step 2: Determine the x and y intercepts
To find the x-intercept, set \( h(x)=0 \) giving \( -x+4/x=0 \). Solving the equation will give \( x=-2 , 2 \) as the x-intercepts. The y- intercept is the value of the function when \( x=0 \). However, for this function, \( x=0 \) is undefined. Therefore, there is no y-intercept.
3Step 3: Identify the asymtotes
Vertical asymptotes occur where the denominator equals zero and the expression is undefined. In this case, it is when \( x=0 \). Horizontal asymptotes occur when the degree of the denominator is equal to or greater than the degree of the numerator. But in this case, the numerator is a linear function and the denominator is a constant giving the slant asymptote. To find the slant asymptote divide the degree of the numerator by the degree of the denominator to get \( y=-x \). Hence the slant asymptote is \( y=-x \)
4Step 4: Sketch the graph
Plot the x-intercepts, y-intercept and asymptote, then sketch the curve. The graph will be approaching the line \( y=-x \) as \( x \) goes to positive and negative infinity, has X-intercepts at -2 and 2, a vertical asymptote at x=0, and no Y-intercept.
Key Concepts
AsymptotesX-InterceptsY-InterceptsSlant Asymptote
Asymptotes
In rational functions, asymptotes are lines that the graph of the function will approach but never touch. These can be vertical, horizontal, or slant.
Vertical asymptotes occur when the denominator of the function equals zero because the function becomes undefined there. In this exercise, because the denominator is simply an \( x \), which equals zero at \( x = 0 \), a vertical asymptote is present at this point.
It is important to always check the denominator of a rational function to understand where the graph cannot touch the line. Consequently, for unconventional asymptotes like slant ones, we'll explore more in the designated section for a clearer understanding.
Vertical asymptotes occur when the denominator of the function equals zero because the function becomes undefined there. In this exercise, because the denominator is simply an \( x \), which equals zero at \( x = 0 \), a vertical asymptote is present at this point.
It is important to always check the denominator of a rational function to understand where the graph cannot touch the line. Consequently, for unconventional asymptotes like slant ones, we'll explore more in the designated section for a clearer understanding.
X-Intercepts
X-intercepts are the points where the graph crosses the x-axis, meaning the output of the function, \( h(x) \), is zero. To find the x-intercepts, set the rational function equal to zero and solve for \( x \).
In this exercise, we have the rational function simplified to \( h(x) = -x + \frac{4}{x} \). Setting it equal to zero, we solve for \( x \) and find that:
In this exercise, we have the rational function simplified to \( h(x) = -x + \frac{4}{x} \). Setting it equal to zero, we solve for \( x \) and find that:
- \( x = 2 \)
- \( x = -2 \)
Y-Intercepts
Y-intercepts are the points where the graph crosses the y-axis, which is where \( x = 0 \). For every rational function, the y-intercept is found by substituting \( x = 0 \) into the function. However, there are exceptions.
In this scenario, plugging \( x = 0 \) into the function \( h(x) = \frac{4-x^{2}}{x} \) results in an undefined value because the denominator becomes zero. Thus, this function has no y-intercept.
When a function lacks a y-intercept, it is often due to a vertical asymptote at that point. So, it's crucial to pay attention to vertical lines of undefined behavior when determining y-intercepts.
In this scenario, plugging \( x = 0 \) into the function \( h(x) = \frac{4-x^{2}}{x} \) results in an undefined value because the denominator becomes zero. Thus, this function has no y-intercept.
When a function lacks a y-intercept, it is often due to a vertical asymptote at that point. So, it's crucial to pay attention to vertical lines of undefined behavior when determining y-intercepts.
Slant Asymptote
Slant asymptotes occur when a rational function's degree in the numerator is exactly one higher than the degree in the denominator, resulting in an asymptote that is neither horizontal nor vertical but rather slanted.
In the given function \( h(x) = \frac{4-x^{2}}{x} \), this condition is met because the numerator (degree of 2) is one degree higher than the denominator (degree of 1). We can find the slant asymptote by performing polynomial long division or simple algebraic manipulation, showing that as the division gives \( y = -x \).
This slant line demonstrates how the function behaves as \( x \) approaches positive or negative infinity.
In the given function \( h(x) = \frac{4-x^{2}}{x} \), this condition is met because the numerator (degree of 2) is one degree higher than the denominator (degree of 1). We can find the slant asymptote by performing polynomial long division or simple algebraic manipulation, showing that as the division gives \( y = -x \).
This slant line demonstrates how the function behaves as \( x \) approaches positive or negative infinity.
Other exercises in this chapter
Problem 56
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