To find the standard form of the quadratic equation given its roots and a point through which the function passes, we need to follow these steps:
1. Identify the roots of the quadratic equation:
The roots given are [tex]\( x = 3 \)[/tex] and [tex]\( x = -1 \)[/tex].
2. Form the quadratic equation from its roots:
The quadratic equation can be expressed as:
[tex]\[
(x - 3)(x + 1)
\][/tex]
Expanding this, we get:
[tex]\[
x^2 + x - 3x - 3 = x^2 - 2x - 3
\][/tex]
3. Include the leading coefficient [tex]\( a \)[/tex] since the quadratic could have a different scaling factor:
The general form of the quadratic equation becomes:
[tex]\[
y = a(x^2 - 2x - 3)
\][/tex]
Here, [tex]\( a \)[/tex] is a constant that we'll determine using the given point [tex]\((1, -10)\)[/tex].
4. Use the given point [tex]\((1, -10)\)[/tex] to find [tex]\( a \)[/tex]:
Substitute [tex]\( x = 1 \)[/tex] and [tex]\( y = -10 \)[/tex] into the equation:
[tex]\[
-10 = a(1^2 - 2(1) - 3)
\][/tex]
Simplifying inside the parentheses:
[tex]\[
-10 = a(1 - 2 - 3)
\][/tex]
[tex]\[
-10 = a(-4)
\][/tex]
Solving for [tex]\( a \)[/tex]:
[tex]\[
a = \frac{-10}{-4} = 2.5
\][/tex]
5. Substitute [tex]\( a \)[/tex] back into the quadratic equation:
[tex]\[
y = 2.5(x^2 - 2x - 3)
\][/tex]
Distributing the [tex]\( a = 2.5 \)[/tex]:
[tex]\[
y = 2.5x^2 - 5x - 7.5
\][/tex]
Thus, the standard form of the quadratic equation is:
[tex]\[
y = 2.5x^2 - 5x - 7.5
\][/tex]
Therefore, the correct answer is:
[tex]\[ y = 2.5 x^2 - 5 x - 7.5 \][/tex]
