To find the standard form of a quadratic function given the roots and a point it passes through, we follow these steps:
1. Identify the given roots: The roots of the quadratic function are [tex]\( r_1 = 2 \)[/tex] and [tex]\( r_2 = -5 \)[/tex].
2. Form the quadratic equation: The quadratic can be expressed as:
[tex]\[
y = a (x - r_1)(x - r_2)
\][/tex]
Substituting the given roots, we have:
[tex]\[
y = a (x - 2)(x + 5)
\][/tex]
3. Convert to standard form:
Expand the expression inside the parentheses:
[tex]\[
y = a (x^2 + 5x - 2x - 10)
\][/tex]
Simplify inside the parentheses:
[tex]\[
y = a (x^2 + 3x - 10)
\][/tex]
4. Use the given point [tex]\((1, -3)\)[/tex] to find [tex]\(a\)[/tex]:
Substitute [tex]\(x = 1\)[/tex] and [tex]\(y = -3\)[/tex] into the equation:
[tex]\[
-3 = a (1^2 + 3(1) - 10)
\][/tex]
Simplify the equation:
[tex]\[
-3 = a (1 + 3 - 10)
\][/tex]
[tex]\[
-3 = a (-6)
\][/tex]
Solving for [tex]\(a\)[/tex]:
[tex]\[
a = \frac{-3}{-6} = 0.5
\][/tex]
5. Form the final equation:
Substitute [tex]\(a = 0.5\)[/tex] back into the expanded form of the equation:
[tex]\[
y = 0.5 (x^2 + 3x - 10)
\][/tex]
6. Distribute [tex]\(0.5\)[/tex]:
[tex]\[
y = 0.5x^2 + 1.5x - 5
\][/tex]
Therefore, the standard form of the quadratic function is:
[tex]\[
y = 0.5x^2 + 1.5x - 5
\][/tex]
Among the given options, the correct equation is:
[tex]\[ y = 0.5x^2 + 1.5x - 5 \][/tex]
