To construct a rational function with the given characteristics, we'll proceed step by step:

1. Vertical Asymptotes at $x = 5$ and $x = -2$:

Vertical asymptotes occur when the denominator of a rational function equals zero. Therefore, the denominator must include the factors $(x - 5)$ and $(x + 2)$, since the function will be undefined at these points.

Thus, the denominator will be: $(x - 5)(x + 2)$

2. x-Intercepts at $x = -5$ and $x = -3$:

x-intercepts occur when the numerator of a rational function equals zero. To achieve intercepts at $x = -5$ and $x = -3$, the numerator must include the factors $(x + 5)$ and $(x + 3)$, as these will result in zero when $x = -5$ and $x = -3$.

Thus, the numerator will be: $(x + 5)(x + 3)$

3. y-Intercept at $y = 7$:

The y-intercept occurs when $x = 0$. To ensure that the function passes through the point $(0, 7)$, we need to scale the rational function by a constant $A$. To find $A$, we substitute $x = 0$ into the equation and set the output to 7.

The function (with the constant $A$) so far is: $f(x) = A \frac{(x + 5)(x + 3)}{(x - 5)(x + 2)}$

Substitute $x = 0$ into the equation: $f(0) = A \frac{(0 + 5)(0 + 3)}{(0 - 5)(0 + 2)} = A \frac{5 \cdot 3}{-5 \cdot 2} = A \frac{15}{-10} = -\frac{3}{2}A$ Since the y-intercept is 7, we set $f(0) = 7$: $-\frac{3}{2}A = 7$ Solve for $A$: $A = -\frac{14}{3}$

4. Final Equation:

Now that we know the constant $A = -\frac{14}{3}$, the equation of the rational function is: $f(x) = -\frac{14}{3} \frac{(x + 5)(x + 3)}{(x - 5)(x + 2)}$

This is the rational function that satisfies all the given conditions.

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Would you like further details on this, or do you have any questions?

Here are 5 related questions for further exploration:

1. How does changing the constant $A$ affect the graph of the rational function?
2. How would the function change if the y-intercept were different?
3. What happens if a rational function has no vertical asymptotes?
4. How do you find horizontal asymptotes of a rational function?
5. What is the significance of a hole in the graph of a rational function?

Tip: Always verify the behavior at intercepts and asymptotes when constructing or analyzing rational functions.