91Ó°ÊÓ

Horizontal intercepts are points where the graph crosses the x-axis.
To find these intercepts for the function given, set the function equal to zero.
Here are the steps for our function, \(h(x)=\frac{1}{(x+3)(x-1)}-2\):
1. Set the function to zero and solve: \(\frac{1}{(x+3)(x-1)}-2=0\) 2. Add 2 to both sides: \(\frac{1}{(x+3)(x-1)}=2\)
3. Take reciprocals: \(1=2(x+3)(x-1)\)
4. Simplify and solve: \(1=2x^2+4x-6\), resulting in \(2x^2+4x-7=0\)
Here, we use the quadratic formula: \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\).
Given \(a=2\), \(b=4\), and \(c=-7\):
\(x=\frac{-4\pm\sqrt{16+56}}{4}\)
Simplify to get: \(x=-1\pm\frac{3\sqrt{2}}{2}\).
These are the horizontal intercepts.

Vertical asymptotes are lines where the function becomes undefined and the graph heads to infinity.
To find these asymptotes, identify where the denominator equals zero.
For our function \(h(x)=\frac{1}{(x+3)(x-1)}-2\):
1. Set the denominator to zero: \((x+3)(x-1)=0\)
2. Solve for \(x\):
\(x=-3\) and \(x=1\)
These solutions indicate vertical asymptotes at \(x=-3\) and \(x=1\).
The graph will approach but never touch these lines.

The quadratic formula is used to solve quadratic equations of the form \(ax^2 + bx + c = 0\).
The formula is:
\(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\).
In our provided problem, we reached a quadratic during our search for horizontal intercepts:
1. After simplifying, we had \(2x^2+4x-7=0\).
2. Here, \(a=2\), \(b=4\), and \(c=-7\).
3. Plug these into the formula to compute:
\(x=\frac{-4\pm\sqrt{16+56}}{4}\)
This simplifies to \(x=-1\pm\frac{3\sqrt{2}}{2}\).
Mastery of the quadratic formula assists in solving many algebraic problems like this.

Graphing technology is a valuable tool for visualizing functions and verifying solutions.
By graphing \(h(x)=\frac{1}{(x+3)(x-1)}-2\), you can observe important features like:
* Horizontal intercepts
* Vertical asymptotes
Software like Desmos or graphing calculators are user-friendly options.
Steps to graph using such technology:
1. Enter the function exactly as given.
2. Look for points where the graph crosses the x-axis to verify horizontal intercepts.
3. Identify vertical lines where the graph does not exist (vertical asymptotes).
Using these tools confirms the intercepts at \(x=-1\pm\frac{3\sqrt{2}}{2}\) and asymptotes at \(x=-3\) and \(x=1\).
Graphing technology not only confirms your calculations but enhances your understanding.