Answer:
Graph of the function touches x-axis twice at 3 and -2.
Step-by-step explanation:
Given : Quadratic function [tex]y= -x^2+x+6[/tex]
To find : How many times does the graph of the function given intersect or touch the x-axis?
Solution :
To find the points graph touch the x-axis i.e, the functions real roots.
Using quadratic formula,
General form - [tex]ax^2+bx+c=0[/tex]
[tex]D=b^2-4ac[/tex]
Solution is [tex]x=\frac{-b\pm\sqrt{D}}{2a}[/tex]
Equation is [tex]y= -x^2+x+6[/tex]
where, a=-1 , b=1, c=6
[tex]D=b^2-4ac[/tex]
[tex]D=(1)^2-4(-1)(6)[/tex]
[tex]D=1+24[/tex]
[tex]D=25[/tex]
D>0 i.e, 2 real roots exist.
Solution is [tex]x=\frac{-b\pm\sqrt{D}}{2a}[/tex]
[tex]x=\frac{-(1)\pm\sqrt{25}}{2(1)}[/tex]
[tex]x=\frac{1\pm 5}{2}[/tex]
[tex]x=3,-2[/tex]
Therefore, graph of the function touches x-axis twice at 3 and -2.
Refer the attached figure below.
