Question

question
click on the buttons below to plot all of the existing features of the following rational function (some may not be needed). click a second time to remove the feature. if you get a fraction or decimal then plot as close to the true location as possible.
$$ f ( x ) = \frac { ( 4 x ^ { 2 } + 20 x + 25 ) ( x + 4 ) } { ( 2 x + 6 ) ( x + 4 ) } $$
plot rational function
vertical asymptote horizontal asymptote slant asymptote
x - intercept y - intercept hole
click on a feature to place it.

Explanation:

Step1: Simplify the function

First, factor \(4x^{2}+20x + 25=(2x + 5)^{2}\) and \(2x+6 = 2(x + 3)\).
The function \(f(x)=\frac{(4x^{2}+20x + 25)(x + 4)}{(2x + 6)(x + 4)}=\frac{(2x + 5)^{2}(x + 4)}{2(x + 3)(x + 4)}\).
Cancel out the common factor \((x + 4)\) (for \(x
eq-4\)), so \(f(x)=\frac{(2x + 5)^{2}}{2(x + 3)}\), \(x
eq-4\).

Step2: Find the hole

Since we canceled out the factor \((x + 4)\), the hole occurs at \(x=-4\).
Substitute \(x=-4\) into the simplified function (after canceling):
\(y=\frac{(2\times(-4)+5)^{2}}{2\times(-4 + 3)}=\frac{(-8 + 5)^{2}}{2\times(-1)}=\frac{9}{-2}=-4.5\). So the hole is at the point \((-4,-4.5)\).

Step3: Find the vertical asymptote

Set the denominator of the simplified function \(2(x + 3)=0\), solve for \(x\).
\(x+3=0\), so \(x=-3\) is the vertical asymptote.

Step4: Find the \(x -\)intercept

Set the numerator of the simplified function \((2x + 5)^{2}=0\).
\(2x+5 = 0\), then \(x=-\frac{5}{2}=-2.5\). So the \(x -\)intercept is at the point \((-2.5,0)\).

Step5: Find the \(y -\)intercept

Set \(x = 0\) in the simplified function \(y=\frac{(2\times0+5)^{2}}{2\times(0 + 3)}=\frac{25}{6}\approx4.17\). So the \(y -\)intercept is at the point \((0,\frac{25}{6})\).

Step6: Determine the type of asymptote (horizontal or slant)

Since the degree of the numerator \(n = 2\) and the degree of the denominator \(m=1\) (\(n>m\)), we use polynomial long - division.
Divide \((2x + 5)^{2}=4x^{2}+20x + 25\) by \(2(x + 3)=2x+6\).
\(4x^{2}+20x + 25=(2x + 6)(2x+4)-(- 1)\). So \(y = 2x+4\) is the slant asymptote.

Answer:

• Hole: \((-4,-4.5)\)
• Vertical Asymptote: \(x=-3\)
• \(x -\)intercept: \((-2.5,0)\)
• \(y -\)intercept: \((0,\frac{25}{6})\)
• Slant Asymptote: \(y = 2x+4\)