To find the quadratic function in the form $f(x) = a(x - r_1)(x - r_2)$, where $a = 4$ and the x-intercepts $r_1$ and $r_2$ are -4 and 1, follow these steps:

1. Start with the formula $f(x) = a(x - r_1)(x - r_2)$.
2. Substitute the values:
   • $a = 4$
   • $r_1 = -4$
   • $r_2 = 1$

So, the function becomes: $f(x) = 4(x - (-4))(x - 1)$ Simplify: $f(x) = 4(x + 4)(x - 1)$

3. Now, expand the expression: $f(x) = 4[(x + 4)(x - 1)]$ Use the distributive property (FOIL): $(x + 4)(x - 1) = x^2 - x + 4x - 4 = x^2 + 3x - 4$

4. Finally, distribute the coefficient $4$: $f(x) = 4(x^2 + 3x - 4)$ $f(x) = 4x^2 + 12x - 16$

Thus, the quadratic function is: $f(x) = 4x^2 + 12x - 16$

Would you like more details or have any questions?

Here are 5 related questions to expand your knowledge:

1. How do the values of $a$, $r_1$, and $r_2$ affect the shape and position of the quadratic function?
2. What happens to the graph of a quadratic function when $a$ is negative?
3. How do you find the vertex of a quadratic function given in standard form?
4. Can you express a quadratic function in vertex form and relate it to standard form?
5. What is the significance of the discriminant in determining the nature of the roots?

Tip: Always double-check by expanding your final expression to verify your function is correct!