Answer:
Step-by-step explanation:
To sketch a quadratic function we need two things:
1) Nature of the curve
2) Vertex
3) y-intercept
The completing square form of the quadratic equation is:
[tex]y=a(x-h)^2+k[/tex]
where,
a represents the nature of the graph it can be maximum or minimum , meaning, if a > 0 then minimum(u shaped curve/happy face) and if a < 0 then maximum(n shaped curve/sad face).
h represents the x-coordinate of the vertex.
k represents the y-coordinate of the vertex.
Now if we compare g(x) with our completing square form we get the following:
[tex]g(x)=(x-4)^2+12\\y=a(x-h)^2+k\\[/tex]
When we simply compare the following we get ,
a = 1 , which means a > 0 since 1 is greater than 0 the nature of the curve will be minimum(happy face/u shaped)
h = 4, which means the x-coordinate of the vertex is 4
k = 12, which means the y-coordinate of the vertex is 12
Now we have the nature of the curve, we have the vertex now all we need is the y-intercept.
For y-intercept:
For y-intercept meaning at which point will the graph cross the y-axis(0 , y)
For that we expand the formula and turn it into the standard quadratic equation form by using the formula (a - b)^2
[tex]y=(x-4)^2+12\\y=((x)^2-2(x)(4)+(4)^2)+12\\y=x^2-8x+16+12\\y=x^2-8x+28\\[/tex]
now we compare with the standard quadratic form:
[tex]y=ax^2+bx+c[/tex]
here c is the y-intercept and while comparing we can see that c = 28 ,
so the curve cuts the y-axis at (0 , 28)
So we have all the three things that we need to graph our function.
So we just plot the y-intercept , the vertex , and join the dots. Just a tip draw a dotted line on the x-coordinate of the vertex because the vertex point is also called as a turning point where the graph goes in the opposite direction just like a mirror reflection. I attached 2 images you can check them out. One is handmade(i know i suck at drawing but still xD) , one is sketched by online graphing calculator.
