Problem 56
Question
Sketch a graph of the rational function and find all intercepts and slant asymptotes. Indicate all asymptotes onthe graph. $$h(x)=\frac{-x^{2}}{x-3}$$
Step-by-Step Solution
Verified Answer
The x-intercept is \(x=0\), the y-intercept is \(y=0\), the vertical asymptote is \(x=3\) and the slant asymptote is \(y=-x\). The graph of the function will have these intercepts and asymptotes.
1Step 1: Find the x-intercept
The x-intercept of a function is the point where it crosses the x-axis. This occurs when \(h(x) = 0\). Substitute \(h(x) = 0\) in the equation to get: \(0=\frac{-x^{2}}{x-3}\). Solve for \(x\) to find the x-intercept.
2Step 2: Find the y-intercept
The y-intercept of a function is the point where it crosses the y-axis. This occurs when \(x = 0\). Substitute \(x = 0\) in the equation to find the y-intercept.
3Step 3: Find the Vertical Asymptote
The vertical asymptote of this function occurs when the denominator is equal to zero. So, set \(x-3 = 0\) and solve for \(x\) to find the vertical asymptote.
4Step 4: Find the Slant Asymptote
For rational functions, the slant asymptote can be found by performing long division of the polynomials. Perform long division of \(-x^{2}\) by \(x-3\). The quotient is the equation of the slant asymptote.
5Step 5: Sketch the Graph
Now, use these values to graph the function. Plot the intercepts and draw the asymptotes. The function approaches the asymptotes but never crosses them. Plot a few other points if necessary and sketch the curve.
Key Concepts
X-InterceptY-InterceptVertical AsymptoteSlant Asymptote
X-Intercept
Understanding the x-intercept of a graph is essential for visualizing how a function behaves in relation to the horizontal axis. In mathematical terms, the x-intercept is where the graph of the function crosses or touches the x-axis. Mathematically speaking, it's found where the output value of the function, denoted as h(x), is zero.
To find the x-intercept of the function h(x) = \frac{-x^{2}}{x-3}, we set the numerator of the function to zero since a fraction is only zero when its numerator is zero. Hence, we get \(0 = -x^{2}\), which immediately gives us the x-intercept at x = 0. What this reveals about the function is that it touches the x-axis at the origin, providing a clear starting point for graphing the function.
To find the x-intercept of the function h(x) = \frac{-x^{2}}{x-3}, we set the numerator of the function to zero since a fraction is only zero when its numerator is zero. Hence, we get \(0 = -x^{2}\), which immediately gives us the x-intercept at x = 0. What this reveals about the function is that it touches the x-axis at the origin, providing a clear starting point for graphing the function.
Y-Intercept
Similar to the x-intercept, the y-intercept is where the graph crosses the y-axis. The y-intercept occurs at the point where the input of the function, x, is zero. For rational functions, finding the y-intercept is straightforward; just plug in zero for the value of x in the function.
So, when we find the y-intercept of h(x) = \frac{-x^{2}}{x-3}, we substitute x with 0, resulting in h(0), which simplifies to \(\frac{0}{-3}\), and therefore, the y-intercept is 0. The information that the y-intercept is 0 is pivotal for plotting the initial behavior of the function on a graph.
So, when we find the y-intercept of h(x) = \frac{-x^{2}}{x-3}, we substitute x with 0, resulting in h(0), which simplifies to \(\frac{0}{-3}\), and therefore, the y-intercept is 0. The information that the y-intercept is 0 is pivotal for plotting the initial behavior of the function on a graph.
Vertical Asymptote
The term vertical asymptote might sound complex, but it's a crucial concept in understanding how rational functions behave. Vertical asymptotes are lines that the graph of a function can get infinitely close to, but never actually touch or cross. These are representative of values that are not in the function's domain.
For h(x) = \frac{-x^{2}}{x-3}, the vertical asymptote can be found by determining the values that make the denominator zero, as the function cannot be defined at these points. Setting the denominator equal to zero, \(x-3=0\), we find that x = 3 is where the vertical asymptote lies. When graphing the function, this line at x = 3 will be a barrier that the curve of h(x) approaches but does not cross.
For h(x) = \frac{-x^{2}}{x-3}, the vertical asymptote can be found by determining the values that make the denominator zero, as the function cannot be defined at these points. Setting the denominator equal to zero, \(x-3=0\), we find that x = 3 is where the vertical asymptote lies. When graphing the function, this line at x = 3 will be a barrier that the curve of h(x) approaches but does not cross.
Slant Asymptote
A slant asymptote is a feature seen in the graphs of certain rational functions that don't level off horizontally or vertically but instead follow a diagonal path. This occurs when the degree of the numerator is exactly one higher than the degree of the denominator. To find a slant asymptote, polynomial long division is used to divide the numerator by the denominator.
Dividing \(-x^{2}\) by \(x-3\), we get a quotient that represents the slant asymptote's equation. In the case of our function h(x), the slant asymptote will help guide the end behavior of the graph for large absolute values of x. Unlike horizontal or vertical asymptotes, the curve may cross a slant asymptote but will eventually follow the path of the slant asymptote as x moves towards infinity or negative infinity.
Dividing \(-x^{2}\) by \(x-3\), we get a quotient that represents the slant asymptote's equation. In the case of our function h(x), the slant asymptote will help guide the end behavior of the graph for large absolute values of x. Unlike horizontal or vertical asymptotes, the curve may cross a slant asymptote but will eventually follow the path of the slant asymptote as x moves towards infinity or negative infinity.
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