To write the quadratic equation in Vertex Form given a vertex at [tex]\((-5, 1)\)[/tex] and a point through which the parabola passes [tex]\((-4, 5)\)[/tex], we can follow these steps:
1. Understand the Vertex Form: A quadratic equation in vertex form is given by:
[tex]\[
y = a(x - h)^2 + k
\][/tex]
Here, [tex]\( (h, k) \)[/tex] is the vertex of the parabola, and [tex]\(a\)[/tex] is a coefficient that determines the width and direction of the parabola.
2. Substitute the Vertex Coordinates: Given the vertex [tex]\((-5, 1)\)[/tex], we substitute [tex]\( h = -5 \)[/tex] and [tex]\( k = 1 \)[/tex] into the vertex form equation:
[tex]\[
y = a(x + 5)^2 + 1
\][/tex]
3. Use the Given Point to Find [tex]\(a\)[/tex]: We know the parabola passes through the point [tex]\((-4, 5)\)[/tex]. So, we substitute [tex]\( x = -4 \)[/tex] and [tex]\( y = 5 \)[/tex] into the equation:
[tex]\[
5 = a(-4 + 5)^2 + 1
\][/tex]
4. Simplify and Solve for [tex]\(a\)[/tex]: Calculate the expression:
[tex]\[
5 = a(1)^2 + 1
\][/tex]
[tex]\[
5 = a + 1
\][/tex]
Subtract 1 from both sides:
[tex]\[
4 = a
\][/tex]
So, the value of [tex]\(a\)[/tex] is 4.
5. Write the Complete Equation: Now that we have determined [tex]\( a = 4 \)[/tex], we substitute it back into the vertex form equation:
[tex]\[
y = 4(x + 5)^2 + 1
\][/tex]
So, the quadratic equation in Vertex Form is:
[tex]\[
y = 4(x + 5)^2 + 1
\][/tex]
