To find the equation of the quadratic function [tex]\( p(x) \)[/tex] in vertex form, we follow these steps:
1. Identify the Vertex:
From the provided table, observe the symmetry of the values. The minimum value (-2) at [tex]\( x = 1 \)[/tex] and equal distances from this point suggest that the vertex lies at [tex]\( x = 2 \)[/tex], where [tex]\( p(2) = 1 \)[/tex].
2. Vertex Form:
The vertex form of a quadratic equation is:
[tex]\[
p(x) = a(x - h)^2 + k
\][/tex]
where [tex]\( (h, k) \)[/tex] is the vertex of the parabola.
Here, [tex]\( h = 2 \)[/tex] and [tex]\( k = 1 \)[/tex], so the equation becomes:
[tex]\[
p(x) = a(x - 2)^2 + 1
\][/tex]
3. Determine the Coefficient [tex]\( a \)[/tex]:
We can use another point from the table to find [tex]\( a \)[/tex]. Let's use the point [tex]\( (1, -2) \)[/tex].
Substitute [tex]\( x = 1 \)[/tex] and [tex]\( p(x) = -2 \)[/tex] into the vertex form:
[tex]\[
-2 = a(1 - 2)^2 + 1
\][/tex]
Simplify and solve for [tex]\( a \)[/tex]:
[tex]\[
-2 = a(-1)^2 + 1
\][/tex]
[tex]\[
-2 = a + 1
\][/tex]
[tex]\[
-2 - 1 = a
\][/tex]
[tex]\[
a = -3
\][/tex]
4. Write the Final Equation:
Substitute [tex]\( a = -3 \)[/tex] back into the vertex form:
[tex]\[
p(x) = -3(x - 2)^2 + 1
\][/tex]
Therefore, the equation of [tex]\( p(x) \)[/tex] in vertex form is:
[tex]\[
p(x) = -3(x - 2)^2 + 1
\][/tex]
