Let's determine the equation of the quadratic function in vertex form [tex]\( t(x) = a(x-h)^2 + k \)[/tex] using the given data points.
### Step-by-Step Solution:
1. Identify the Given Points:
The points given are:
[tex]\[
(6, -2), (7, 4), (8, 6), (9, 4), (10, -2)
\][/tex]
2. Fit a Quadratic Function:
For a quadratic function, the general form is [tex]\( f(x) = ax^2 + bx + c \)[/tex].
3. Vertex Form of a Quadratic Function:
The vertex form of a quadratic function is given by:
[tex]\[
t(x) = a(x-h)^2 + k
\][/tex]
where [tex]\( (h, k) \)[/tex] is the vertex of the parabola.
4. Convert Coefficients to Vertex Form:
From the analysis, we determine the coefficients of the quadratic function that fits the points:
[tex]\[
a = -2.0, \quad h = 8.0 \quad (\text{approximately 7.999999999999996}), \quad k = 6.0 \quad (\text{approximately 5.999999999999943})
\][/tex]
5. Substitute the Values:
Using the values of [tex]\( a \)[/tex], [tex]\( h \)[/tex], and [tex]\( k \)[/tex], we can directly write the equation in vertex form:
[tex]\[
t(x) = -2(x - 8)^2 + 6
\][/tex]
### Final Equation:
The equation of the function in vertex form is:
[tex]\[
t(x) = -2(x - 8)^2 + 6
\][/tex]
Thus, we have successfully determined the quadratic function's equation in its vertex form, given the data points in the table.
