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Vertical asymptotes occur in a function where the denominator equals zero, leading to undefined points in the graph. In the function \( f(x) = \frac{x+12}{x-6} \), the denominator is \( x-6 \). Setting this expression to zero, we find that the vertical asymptote is located at \( x = 6 \). This does not just create a hole in the graph but forms a line that the graph approaches but never touches or crosses.

This characteristic tells us that at \( x = 6 \), the values of \( f(x) \) will shoot to positive or negative infinity, depending on the direction of approach.

• The graph approaches \( +\infty \) as \( x \) approaches \( 6 \) from the left.
• It approaches \( -\infty \) as \( x \) approaches \( 6 \) from the right.

Understanding this helps you sketch accurate graphs and recognize behavior patterns near undefined points.

Horizontal asymptotes describe the behavior of a function as \( x \) approaches positive infinity or negative infinity. Unlike vertical asymptotes, which indicate undefined points, horizontal asymptotes suggest a value that the function approaches but rarely touches.

For the function \( f(x) = \frac{x+12}{x-6} \), both the numerator and the denominator are first-degree polynomials (their degrees are equal). This results in a horizontal asymptote. To find it, divide the leading coefficients of the numerator and denominator. This gives us a horizontal asymptote at \( y = 1 \).

• As \( x \to \infty \), \( f(x) \to 1 \).
• As \( x \to -\infty \), \( f(x) \to 1 \).

This asymptote indicates that as the values of \( x \) become very large or very small, the graph of the function will get closer to the line \( y = 1 \). However, it may never actually reach that line, giving it a unique boundary character in the graph's far ends.

Finding intercepts in a function's graph helps us understand where it crosses the axes. These points are crucial for accurately plotting the graph and recognizing the function's roots and behavior.

For the function \( f(x) = \frac{x+12}{x-6} \), let's find the intercepts:

• **X-intercept:** Set the function equal to zero: \( x + 12 = 0 \). Solving this gives \( x = -12 \). Thus, the graph crosses the x-axis at \((-12, 0)\).
• **Y-intercept:** Set \( x = 0 \) in the function: \( f(0) = \frac{12}{-6} = -2 \). Hence, the graph crosses the y-axis at \((0, -2)\).

These intercepts are points where the function intersects the x-axis and y-axis. Recognizing the intercepts provides a framework for sketching the basic shape of the graph more accurately, as it shows where the function changes from positive to negative or vice-versa.