A quadratic function of the form  
f
(
x
)
=
a
x
2
+
b
x
+
c
  with  
b
2
−
4
a
c
>
0
  may also be written in the form  
f
(
x
)
=
a
(
x
−
r
1
)
(
x
−
r
2
)
 , where  
r
1
  and  
r
2
  are the x-intercepts of the graph of the quadratic function. Find a quadratic function whose x-intercepts are - 4 and 1 with  
a
=
4
 .

Provide your answer below:

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!

A quadratic function of the form f(x) = ax^2 + bx + c with b^2 − 4ac > 0 may also be written in the form f(x) = a(x − r1)(x − r2), where r1 and r2 are the x-intercepts of the graph of the quadratic function. Find a quadratic function whose x-intercepts are -4 and 1 with a = 4.

Solve a quadratic function with given x-intercepts of -4 and 1, and a leading coefficient of a = 4. This problem focuses on key concepts of quadratic equations, including converting standard form to factored form and expanding binomials. Suitable for Grades 9-10.

Math Problem Statement

A quadratic function of the form
f ( x )

0 may also be written in the form
f ( x )

a ( x − r 1 ) ( x − r 2 ) , where
r 1 and
r 2 are the x-intercepts of the graph of the quadratic function. Find a quadratic function whose x-intercepts are - 4 and 1 with
a

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