91Ó°ÊÓ

Intercepts in a rational function are points where the graph crosses the axes. This includes the x-intercept and y-intercept. To find the x-intercept of the function \( t(x) = \frac{3x + 6}{x^2 + 2x - 8} \), you set the numerator equal to zero because the fraction only equals zero when its numerator is zero. Solve \( 3x + 6 = 0 \) to find \( x = -2 \). This tells us the x-intercept is \( (-2, 0) \).

For the y-intercept, set \( x = 0 \) in the function. This means calculating \( t(0) = \frac{3(0) + 6}{0^2 + 2(0) - 8} = -\frac{3}{4} \). Hence, the y-intercept is at \( (0, -\frac{3}{4}) \). These points are useful for sketching the initial path of the graph.

Asymptotes are lines that a graph approaches but never actually touches. There are two types to find for rational functions: vertical and horizontal asymptotes.

Vertical asymptotes occur where the denominator is zero, so factor the denominator \( x^2 + 2x - 8 \) to \( (x - 2)(x + 4) \). Set each factor to zero: \( x - 2 = 0 \) and \( x + 4 = 0 \) give \( x = 2 \) and \( x = -4 \) as vertical asymptotes.

To find horizontal asymptotes, compare the degrees of the numerator and the denominator. Here, the numerator has degree 1 and the denominator has degree 2. Since the numerator's degree is less, the horizontal asymptote is \( y = 0 \). The graph will get very close to these lines but will not touch them.

Graphing a rational function like \( t(x) = \frac{3x + 6}{x^2 + 2x - 8} \) requires plotting intercepts and asymptotes. Start by marking the x-intercept \((-2, 0)\) and y-intercept \((0, -\frac{3}{4})\).

Draw vertical dashed lines at \( x = 2 \) and \( x = -4 \) to represent the vertical asymptotes, and a horizontal dashed line along \( y = 0 \) as the horizontal asymptote.

Sketch the curve such that it approaches these asymptotes, but never crosses them. Near vertical asymptotes, the graph will rise or fall sharply. As \( x \) moves towards positive or negative infinity, the graph will flatten out towards \( y = 0 \). Finally, confirm your sketch with a graphing device to ensure accuracy. This process helps visualize how the rational function behaves, providing insights into its behavior across different values of \( x \).