What is the range of the graph of #y = 5(x – 2)^2 + 7#?

1 Answer
May 9, 2018

Answer:

#color(blue)(y in[7,oo)#

Explanation:

Notice #y=5(x-2)^2+7# is in the vertex form of a quadratic:

#y=a(x-h)^2+k#

Where:

#bba# is the coefficient of #x^2#, #bbh# is the axis of symmetry and #bbk# is the maximum/minimum value of the function.

If:

#a>0# then the parabola is of the form #uuu# and #k# is a minimum value.

In example:

#5>0#

#k=7#

so #k# is a minimum value.

We now see what happens as #x->+-oo#:

as #x->oocolor(white)(88888)#, #5(x-2)^2+7->oo#

as #x->-oocolor(white)(888)#, #5(x-2)^2+7->oo#

So the range of the function in interval notation is:

#y in[7,oo)#

This is confirmed by the graph of #y=5(x-2)^2+7#

graph{y=5(x-2)^2+7 [-10, 10, -5, 41.6]}