Question

How many times does the graph of the function below intersect or touch the x-axis? y= -x^2+x+6

Answer 1

2 times, once at 3 and once at -2

Answer 2

Answer:

Graph of the function touches x-axis twice at 3 and -2.

Step-by-step explanation:

Given : Quadratic function $y= -x^2+x+6$

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 - $ax^2+bx+c=0$

$D=b^2-4ac$  

Solution is $x=\frac{-b\pm\sqrt{D}}{2a}$  

Equation is $y= -x^2+x+6$

where, a=-1 , b=1, c=6

$D=b^2-4ac$  

$D=(1)^2-4(-1)(6)$  

$D=1+24$

$D=25$

D>0 i.e, 2 real roots exist.

Solution is $x=\frac{-b\pm\sqrt{D}}{2a}$

$x=\frac{-(1)\pm\sqrt{25}}{2(1)}$  

$x=\frac{1\pm 5}{2}$  

$x=3,-2$  

Therefore, graph of the function touches x-axis twice at 3 and -2.

Refer the attached figure below.