Find the x-intercept(s) and y -intercept(s) of the graph of the following. y=4x+3/x^2+4      If there is more than one answer, separate them with commas.
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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**: \( \left(0, \frac{3}{4}\right) \)

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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.

Find the x-intercept(s) and y-intercept(s) of the graph of the following. y = (4x + 3) / (x^2 + 4). If there is more than one answer, separate them with commas.

Discover how to find the x- and y-intercepts of the rational function y = (4x + 3) / (x^2 + 4). This solution covers the step-by-step method for determining intercepts using algebraic principles. Ideal for students in Grades 10-12.

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