To determine the values of \( a \), \( h \), and \( k \) in the vertex form of the quadratic equation \( f(x) = a(x-h)^2 + k \), following steps were considered:
1. Identify the coefficient \( a \):
The coefficient \( a \) represents the vertical stretch or compression of the parabola along the y-axis. It also determines the direction of the parabola. If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards. In this case, the given coefficient is:
[tex]\[
a = 2
\][/tex]
2. Identify the horizontal shift \( h \):
The value \( h \) represents the x-coordinate of the vertex of the parabola, which translates the graph horizontally. It indicates how much the graph is shifted to the left or right from the origin. Here, the x-coordinate of the vertex is:
[tex]\[
h = 3
\][/tex]
3. Identify the vertical shift \( k \):
The value \( k \) represents the y-coordinate of the vertex of the parabola, which translates the graph vertically. It indicates how much the graph is shifted up or down from the origin. Here, the y-coordinate of the vertex is:
[tex]\[
k = 5
\][/tex]
Hence, the values in the vertex form equation \( f(x) = a(x-h)^2 + k \) are:
[tex]\[
a = 2
\][/tex]
[tex]\[
h = 3
\][/tex]
[tex]\[
k = 5
\][/tex]
