Problem 7
Question
Find the horizontal and vertical asymptotes of the graph of the function defined by the given equation, and draw a sketch of the graph. $$ h(x)=\frac{4 x^{2}}{x^{2}-9} $$
Step-by-Step Solution
Verified Answer
Vertical asymptotes at \ x = \pm 3 \ and horizontal asymptote at \ y = 4 \.
1Step 1 - Find Vertical Asymptotes
Vertical asymptotes occur where the denominator is equal to zero. Set the denominator of the function \(x^{2} - 9 \) equal to zero and solve for \( x \). \ x^{2} - 9 = 0 \rightrightarrow x^{2} = 9 \rightrightarrow x = \pm 3\. So, the vertical asymptotes are \ x = 3 \ and \ x = -3 \.
2Step 2 - Find Horizontal Asymptote
Horizontal asymptotes are found by comparing the degrees of the numerator and denominator. The degree of both the numerator \(4x^{2}\) and the denominator \(x^{2} - 9\) are 2. Since the degrees are the same, the horizontal asymptote is determined by the ratio of the leading coefficients. \h\or\izontal\ \a\s\ymptote: \frac{4}{1} = 4\. So, the horizontal asymptote is \ y = 4 \.
3Step 3 - Draw a Sketch of the Graph
To sketch the graph, draw the vertical asymptotes as dotted lines at \( x = 3 \ and \ x = -3 \). Then, draw the horizontal asymptote as a dotted line at \ y = 4 \. Plot a few points to understand the behaviour of the function around these asymptotes and draw the curve that approaches these lines.
Key Concepts
Vertical AsymptotesHorizontal AsymptotesGraphing Rational FunctionsAsymptote DeterminationFunction Behavior Analysis
Vertical Asymptotes
Vertical asymptotes represent values of x where the function tends to infinity or negative infinity. These occur when the denominator of the function equals zero, making the function undefined.
For the function \(h(x)= \frac{4x^2}{x^2 - 9}\), we set the denominator equal to zero to find the vertical asymptotes.
Solve:
\( x^2 - 9 = 0 \rightarrow x^2 = 9 \rightarrow x = \text{±} 3 \).
Therefore, the vertical asymptotes are at \(x = 3\) and \(x = -3\). This means the graph will approach these lines but never touch or cross them.
For the function \(h(x)= \frac{4x^2}{x^2 - 9}\), we set the denominator equal to zero to find the vertical asymptotes.
Solve:
\( x^2 - 9 = 0 \rightarrow x^2 = 9 \rightarrow x = \text{±} 3 \).
Therefore, the vertical asymptotes are at \(x = 3\) and \(x = -3\). This means the graph will approach these lines but never touch or cross them.
Horizontal Asymptotes
Horizontal asymptotes provide insight into the behavior of a function as x approaches infinity or negative infinity. To determine horizontal asymptotes for rational functions, we compare the degrees of the numerator and the denominator.
In this case, \(h(x)= \frac{4x^2}{x^2 - 9}\), both the numerator and denominator have the same degree, which is 2.
When the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients. Here:
Horizontal Asymptote: \ \frac{4}{1} = 4 \, so the horizontal asymptote is \(y = 4\).
In this case, \(h(x)= \frac{4x^2}{x^2 - 9}\), both the numerator and denominator have the same degree, which is 2.
When the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients. Here:
Horizontal Asymptote: \ \frac{4}{1} = 4 \, so the horizontal asymptote is \(y = 4\).
Graphing Rational Functions
When graphing rational functions, it’s crucial to mark both the vertical and horizontal asymptotes correctly.
Start by plotting the vertical asymptotes as vertical dashed lines at \(x = 3\) and \(x = -3\).
Next, draw the horizontal asymptote as a horizontal dashed line at \(y = 4\).
By identifying these asymptotes, we establish boundaries that the graph cannot cross, which helps in sketching the curve's general shape.
Start by plotting the vertical asymptotes as vertical dashed lines at \(x = 3\) and \(x = -3\).
Next, draw the horizontal asymptote as a horizontal dashed line at \(y = 4\).
By identifying these asymptotes, we establish boundaries that the graph cannot cross, which helps in sketching the curve's general shape.
Asymptote Determination
Determining asymptotes involves both algebraic manipulation and understanding the behavior of rational functions.
- Vertical asymptotes are found by setting the denominator equal to zero and solving for x.
- Horizontal asymptotes are found by comparing the degrees of the numerator and denominator or using their leading coefficients when degrees are equal.
Function Behavior Analysis
Analyzing function behavior around asymptotes helps in understanding the graph more deeply.
Near vertical asymptotes \((x = 3 \text{ and } x = -3)\), the function values tend towards positive or negative infinity based on the numerator's sign and value close to these points.
For horizontal asymptotes \((y = 4)\), observe that as x approaches large positive or negative values, the function \(h(x)\) approaches 4.
This asymptotic behavior helps in predicting the end behavior of the function and sketching a more accurate graph.
Near vertical asymptotes \((x = 3 \text{ and } x = -3)\), the function values tend towards positive or negative infinity based on the numerator's sign and value close to these points.
For horizontal asymptotes \((y = 4)\), observe that as x approaches large positive or negative values, the function \(h(x)\) approaches 4.
This asymptotic behavior helps in predicting the end behavior of the function and sketching a more accurate graph.
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