Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Connect with a community of experts ready to help you find accurate solutions to your questions quickly and efficiently. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
Given:
The given function is:
[tex]f(x)=-(x-2)^2+9[/tex]
To find:
The transformations, intercepts and the vertex.
Solution:
The vertex form of a parabola is:
[tex]y=a(x-h)^2+k[/tex] ...(i)
Where, a is a constant and (h,k) is vertex.
If a<0, then the graph of parent quadratic function [tex]y=x^2[/tex] reflect across the x-axis.
If h<0, then the graph of parent function shifts h units left and if h>0, then the graph of parent function shifts h units right.
If k<0, then the graph of parent function shifts k units down and if k>0, then the graph of parent function shifts k units up.
We have,
[tex]f(x)=-(x-2)^2+9[/tex] ...(ii)
On comparing (i) and (ii), we get
[tex]a=-1,h=2,k=9[/tex]
So, the graph of the parent function reflected across the x-axis, and shifts 2 units right and 9 units up.
Putting x=0 in (ii), we get
[tex]f(0)=-(0-2)^2+9[/tex]
[tex]f(0)=-4+9[/tex]
[tex]f(0)=5[/tex]
The y-intercept is 5.
Putting f(x)=0 in (ii), we get
[tex]0=-(x-2)^2+9[/tex]
[tex](x-2)^2=9[/tex]
Taking square root on both sides, we get
[tex](x-2)=\pm \sqrt{9}[/tex]
[tex]x=\pm 3+2[/tex]
[tex]x=3+2\text{ and }x=-3+2[/tex]
[tex]x=5\text{ and }x=-1[/tex]
Therefore, the x-intercepts are -1 and 5.
The values of h and k are 2 and 9 respectively and (h,k) is the vertex of the parabola.
Therefore, the vertex of the parabola is (2,9).
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.
