Question

How do you use the important points to sketch the graph of `y = x^2 - 2x + 5`?

Answer 1

see explanation.

Explanation:

`" the following points are useful"`

`• " coordinates of vertex"`

`• " x and y intercepts"`

`• " shape- maximum or minimum"`

`color(blue)"coordinates of vertex"`

`"for the standard quadratic function "`

`y=ax^2+bx+c ; a!=0`

`"the x-coordinate of the vertex is"`

`x_(color(red)"vertex")=-b/(2a)`

`"here " a=1,b=-2" and " c=5`

`rArrx_(color(red)"vertex")=-(-2)/2=1`

`"substitute this value into function to obtain y"`

`y_(color(red)"vertex")=1^2-2(1)+5=4`

`rArrcolor(magenta)"vertex " =(1,4)`

`color(blue)"Intercepts"`

`• "let x = 0, in function, for y-intercept"`

`• " let y = 0, in function, for x-intercept"`

`x=0toy=5larrcolor(red)" y-intercept"`

`y=0tox^2-2x+5=0`

`"calculate the value of the "color(blue)"discriminant"`

`b^2-4ac=(-2)^2-(4xx1xx5)=4-20=-16`

`"since discriminant " < 0" there are no x-intercepts"`

`color(blue)"shape of parabola"`

`• " if " a>0" then minimum " uuu`

`• " if " a<0" then maximum " nnn`

`"here " a=1 >0rArr" minimum"`

`"plot the points " (0,5)" and "`(1,4)
`"and draw a smooth curve through them"`
graph{x^2-2x+5 [-10, 10, -5, 5]}