To find the x-intercepts and y-intercepts of the function $y = \frac{4x + 3}{x^2 + 4}$, we need to follow these steps:

1. Find the y-intercept:

The y-intercept occurs when $x = 0$. Substituting $x = 0$ into the equation:

$y = \frac{4(0) + 3}{0^2 + 4} = \frac{3}{4}$

Thus, the y-intercept is $\left(0, \frac{3}{4}\right)$.

2. Find the x-intercept(s):

The x-intercepts occur when $y = 0$. Set the function equal to zero and solve for $x$:

$\frac{4x + 3}{x^2 + 4} = 0$

For a fraction to be zero, the numerator must be zero. So, solve:

$4x + 3 = 0$

$4x = -3$

$x = -\frac{3}{4}$

Thus, the x-intercept is $\left(-\frac{3}{4}, 0\right)$.

Final answer:

• x-intercept: $\left(-\frac{3}{4}, 0\right)$
• y-intercept: $\left(0, \frac{3}{4}\right)$

Would you like further details on any part of this solution?

Here are five related questions:

1. How do you find intercepts for a quadratic function?
2. Can a function have multiple x- or y-intercepts?
3. What does the behavior of the graph near the x- and y-intercepts tell us?
4. How do vertical asymptotes affect intercepts?
5. How would the intercepts change if the function had different terms?

Tip: The x-intercepts of a rational function depend only on when the numerator equals zero, not the denominator.