Answer :
To graph the function [tex]\( f(x) = \frac{x(x-4)(x+4)}{x^2 - 8x + 12} \)[/tex], we need to consider several important steps and characteristics of the function.
### 1. Simplify the Function
First, let's simplify the expression where possible:
[tex]\[ f(x) = \frac{x(x-4)(x+4)}{x^2 - 8x + 12} \][/tex]
Factor the denominator:
[tex]\[ x^2 - 8x + 12 = (x-6)(x-2) \][/tex]
So the function becomes:
[tex]\[ f(x) = \frac{x(x-4)(x+4)}{(x-6)(x-2)} \][/tex]
### 2. Domain of the Function
The function is undefined where the denominator is zero:
[tex]\[ x^2 - 8x + 12 = 0 \][/tex]
[tex]\[ (x-6)(x-2) = 0 \][/tex]
So, [tex]\( x = 6 \)[/tex] and [tex]\( x = 2 \)[/tex] will make the denominator zero. Therefore, the domain of the function is all real numbers except [tex]\( x = 6 \)[/tex] and [tex]\( x = 2 \)[/tex].
### 3. Simplify the Function Further
Before simplifying further, recognize that the numerator and denominator do not share common factors. Thus, the simplified form remains as:
[tex]\[ f(x) = \frac{x(x-4)(x+4)}{(x-6)(x-2)} \][/tex]
### 4. Analyze the Asymptotes and Discontinuities
- Vertical Asymptotes: These occur where the function is undefined, which are [tex]\( x = 6 \)[/tex] and [tex]\( x = 2 \)[/tex].
- Horizontal Asymptotes: To find horizontal asymptotes, consider the degrees of the numerator and denominator. Since the numerator is a cubic polynomial [tex]\( x^3 \)[/tex] and the denominator is a quadratic polynomial [tex]\( x^2 \)[/tex], as [tex]\( x \)[/tex] tends to infinity, the function will tend towards infinity as well. This means there is no horizontal asymptote.
### 5. Find Intercepts
- x-intercepts: These are found by setting the numerator equal to zero.
[tex]\[ x(x-4)(x+4) = 0 \][/tex]
So, [tex]\( x = 0, x = 4, x = -4 \)[/tex]. Thus, the x-intercepts are at (0,0), (4,0), and (-4,0).
- y-intercepts: This is found by evaluating f(x) at [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = \frac{0(0-4)(0+4)}{(0-6)(0-2)} = 0 \][/tex]
So the y-intercept is at (0,0).
### 6. Sketch the Graph
Based on the above information, we can now sketch the graph:
1. Draw the coordinate axes.
2. Mark the discontinuities at [tex]\( x = 6 \)[/tex] and [tex]\( x = 2 \)[/tex] with vertical dashed lines to indicate vertical asymptotes.
3. Plot the intercepts at [tex]\( (0,0) \)[/tex], [tex]\( (4,0) \)[/tex], and [tex]\( (-4,0) \)[/tex].
4. Draw the curve passing through these intercepts, considering the behaviour near the vertical asymptotes and the general increasing/decreasing pattern based on polynomial characteristics.
It is essential to remember that near the vertical asymptotes, the function will tend towards [tex]\( \pm \infty \)[/tex] depending upon the direction from which it approaches the asymptotes.
### Conclusion
The function [tex]\( f(x) \)[/tex] has a complex behaviour due to its rational form with asymptotes and intercepts. By following each of these steps, a careful sketch of the function can be constructed, showing how it behaves across its domain.
### 1. Simplify the Function
First, let's simplify the expression where possible:
[tex]\[ f(x) = \frac{x(x-4)(x+4)}{x^2 - 8x + 12} \][/tex]
Factor the denominator:
[tex]\[ x^2 - 8x + 12 = (x-6)(x-2) \][/tex]
So the function becomes:
[tex]\[ f(x) = \frac{x(x-4)(x+4)}{(x-6)(x-2)} \][/tex]
### 2. Domain of the Function
The function is undefined where the denominator is zero:
[tex]\[ x^2 - 8x + 12 = 0 \][/tex]
[tex]\[ (x-6)(x-2) = 0 \][/tex]
So, [tex]\( x = 6 \)[/tex] and [tex]\( x = 2 \)[/tex] will make the denominator zero. Therefore, the domain of the function is all real numbers except [tex]\( x = 6 \)[/tex] and [tex]\( x = 2 \)[/tex].
### 3. Simplify the Function Further
Before simplifying further, recognize that the numerator and denominator do not share common factors. Thus, the simplified form remains as:
[tex]\[ f(x) = \frac{x(x-4)(x+4)}{(x-6)(x-2)} \][/tex]
### 4. Analyze the Asymptotes and Discontinuities
- Vertical Asymptotes: These occur where the function is undefined, which are [tex]\( x = 6 \)[/tex] and [tex]\( x = 2 \)[/tex].
- Horizontal Asymptotes: To find horizontal asymptotes, consider the degrees of the numerator and denominator. Since the numerator is a cubic polynomial [tex]\( x^3 \)[/tex] and the denominator is a quadratic polynomial [tex]\( x^2 \)[/tex], as [tex]\( x \)[/tex] tends to infinity, the function will tend towards infinity as well. This means there is no horizontal asymptote.
### 5. Find Intercepts
- x-intercepts: These are found by setting the numerator equal to zero.
[tex]\[ x(x-4)(x+4) = 0 \][/tex]
So, [tex]\( x = 0, x = 4, x = -4 \)[/tex]. Thus, the x-intercepts are at (0,0), (4,0), and (-4,0).
- y-intercepts: This is found by evaluating f(x) at [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = \frac{0(0-4)(0+4)}{(0-6)(0-2)} = 0 \][/tex]
So the y-intercept is at (0,0).
### 6. Sketch the Graph
Based on the above information, we can now sketch the graph:
1. Draw the coordinate axes.
2. Mark the discontinuities at [tex]\( x = 6 \)[/tex] and [tex]\( x = 2 \)[/tex] with vertical dashed lines to indicate vertical asymptotes.
3. Plot the intercepts at [tex]\( (0,0) \)[/tex], [tex]\( (4,0) \)[/tex], and [tex]\( (-4,0) \)[/tex].
4. Draw the curve passing through these intercepts, considering the behaviour near the vertical asymptotes and the general increasing/decreasing pattern based on polynomial characteristics.
It is essential to remember that near the vertical asymptotes, the function will tend towards [tex]\( \pm \infty \)[/tex] depending upon the direction from which it approaches the asymptotes.
### Conclusion
The function [tex]\( f(x) \)[/tex] has a complex behaviour due to its rational form with asymptotes and intercepts. By following each of these steps, a careful sketch of the function can be constructed, showing how it behaves across its domain.
