Given the function $y = \frac{ax + b}{cx + d}$, where $a$, $b$, $c$, and $d$ are constants and $c \neq 0$, we need to find the intercepts with the axes.

1. x-intercept:

The x-intercept occurs when $y = 0$.

Set the function equal to zero: $0 = \frac{ax + b}{cx + d}$

For a fraction to be zero, the numerator must be zero: $ax + b = 0$

Solve for $x$: $x = -\frac{b}{a}$ So, the x-intercept is at $x = -\frac{b}{a}$.

2. y-intercept:

The y-intercept occurs when $x = 0$.

Substitute $x = 0$ into the function: $y = \frac{a(0) + b}{c(0) + d} = \frac{b}{d}$

So, the y-intercept is at $y = \frac{b}{d}$.

Summary:

• x-intercept: $x = -\frac{b}{a}$, or the point $\left(-\frac{b}{a}, 0\right)$
• y-intercept: $y = \frac{b}{d}$, or the point $\left(0, \frac{b}{d}\right)$

Would you like more details or have any further questions?

Related Questions:

1. How do the values of $a$, $b$, $c$, and $d$ affect the shape of the graph?
2. What happens if $a = 0$ or $b = 0$?
3. Can you find the vertical asymptote for this function?
4. How does the sign of $c$ impact the graph?
5. What is the behavior of the function as $x$ approaches infinity?

Tip: Always check for undefined points in rational functions by analyzing where the denominator equals zero.