Answer:
A. [tex]y=x^2+9x+18[/tex]
Step-by-step explanation:
By the given graph,
The function is intersecting x-axis at - 3 and - 6,
Thus, the x-intercepts of the given function must be (-3,0) and (-6,0),
Now, In option A,
The function is,
[tex]y=x^2+9x+18[/tex]
For x-intercept y = 0,
[tex]x^2+9x+18=0[/tex]
[tex]x^2+6x+3x+18=0[/tex]
[tex]x(x+6)+3(x+6)=0[/tex]
[tex](x+3)(x+6)=0[/tex]
[tex]\implies x = -3\text{ or }x = -6[/tex]
⇒ x-intercepts are (-3,0), (-6,0)
⇒ The function [tex]y=x^2+9x+18[/tex] describes the given graph,
Now, In option B,
The function is,
[tex] y=(x-3)(x-6)[/tex]
For x-intercept y = 0,
[tex](x-3)(x-6)=0[/tex]
[tex]\implies x = 3\text{ or }x = 6[/tex]
⇒ x-intercepts are (3,0), (6,0)
⇒ The function [tex] y=(x-3)(x-6)[/tex] does not describe the given graph,
Now, In option C,
The function is,
[tex]y=x^2-2x+4[/tex]
For x-intercept y = 0,
[tex]y=x^2-2x+4=0[/tex]
[tex](x-2)^2=0[/tex]
[tex]\implies x = 2[/tex]
⇒ x-intercept is (2,0),
⇒ The function [tex]y=x^2-2x+4[/tex] doesn't describe the given graph,
Now, In option D,
The function is,
[tex]y=(x+5)(x-4)[/tex]
For x-intercept y = 0,
[tex](x+5)(x-4)=0[/tex]
[tex]\implies x = -5\text{ or }x = 4[/tex]
⇒ x-intercepts are (-5,0), (4,0)
⇒ The function [tex]y=(x+5)(x-4)[/tex] does not describe the given graph.