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Intercepts are points where the graph of a function crosses the axes. There are two types of intercepts in a rational function, the x-intercept and the y-intercept. Understanding how to find these points is essential for sketching the graph of a rational function. To find the **y-intercept** of a function, substitute 0 for every instance of x in the equation. By doing this in the function \(s(x) = \frac{4 - 3x}{x+7}\), you find:

• \(s(0) = \frac{4}{7}\)

So, the y-intercept is at the point \((0, \frac{4}{7})\). This is where the graph touches or crosses the y-axis.

For the **x-intercept**, set the whole function equal to zero and solve for x. This occurs when the numerator is zero because a rational function is zero when the numerator is zero and the denominator is not. Solving for \(x\) in:

• \(4 - 3x = 0\)
• \(3x = 4\)
• \(x = \frac{4}{3}\)

This gives the x-intercept at \((\frac{4}{3}, 0)\). This is where the graph touches or crosses the x-axis.

Asymptotes are lines that the graph of a function approaches but never actually touches. They help us understand the behavior of the function as it goes to infinity. Rational functions often have two types of asymptotes: vertical and horizontal.**Vertical asymptotes** occur at values of x that make the denominator zero, as long as the numerator is not zero at these points. They represent the values that x cannot take in the function's domain. For the function \(s(x) = \frac{4 - 3x}{x+7}\), setting the denominator equal to zero:

• \(x + 7 = 0\)
• \(x = -7\)

This indicates a vertical asymptote at \(x = -7\). The graph approaches this line but does not cross it.

**Horizontal asymptotes** provide information about the behavior of the function as x approaches infinity. For rational functions, the horizontal asymptote can be found by comparing the degrees of the numerator and the denominator.

• If they are equal, the horizontal asymptote is the ratio of the leading coefficients.
• For \(s(x)\), both numerator and denominator are of degree 1; hence:
• \(y = \frac{-3}{1} = -3\)

So, \(y = -3\) is the horizontal asymptote, showing the function's end behavior.

Graphing rational functions combines everything you've learned about intercepts and asymptotes. Start by plotting the x and y intercepts. These give anchor points for your graph on the axes. In our example, these are \((0, \frac{4}{7})\) and \((\frac{4}{3}, 0)\), respectively.

Next, integrate the asymptotes into your graph. Draw a vertical line at \(x = -7\) and a horizontal line at \(y = -3\). These lines set boundaries that your graph will approach but never cross.
As you sketch the curve, remember:

• The graph will get very close to the asymptotes without actually touching them.
• Intercept points should be included and clearly marked on the graph.
• Consider how the function behaves between and beyond these intercepts and asymptotes.

Finally, use graphing tools to confirm your hand-drawn sketch. Enter \(s(x) = \frac{4 - 3x}{x + 7}\) into a graphing calculator. This visual confirmation ensures your understanding and identifies any potential errors in manual graphing.