The equation of a parabola in vertex form is:
[tex]f(x)=a(x-h)^2+k[/tex]
Where (h,k) are the coordinates of the vertex of the parabola.
From the picture, notice that the vertex of the parabola has coordinates (3,23), then:
[tex]\begin{gathered} h=3 \\ k=23 \end{gathered}[/tex]
Then, our function reduces to:
[tex]f(x)=a(x-3)^2+23[/tex]
Evaluate the function at x=0 to find the value of a. Notice from the figure that f(0)=14. On the other hand:
[tex]\begin{gathered} f(0)=a(0-3)^2+23 \\ \Rightarrow f(0)=a\cdot9+23 \\ \Rightarrow f(0)=9a+23 \end{gathered}[/tex]
Since f(0)=14 then:
[tex]\begin{gathered} 14=9a+23 \\ \Rightarrow9a=14-23 \\ \Rightarrow9a=-9 \\ \Rightarrow a=-\frac{9}{9} \\ \therefore a=-1 \end{gathered}[/tex]
Therefore, the function that describes this situation is:
[tex]f(x)=-(x-3)^2+23[/tex]
