What is the range of #f(x)=25x^2-30x+16#?

1 Answer
Nov 15, 2015

Answer:

#y>=1129/144#

Explanation:

The range of a function is the set of all the y-coordinates that are represented in the function. A good way to tell range can be simply through looking at a graph.
graph{25x^2-30x+16 [-15.19, 16.85, -2.13, 15.56]}
This is the graph of #f(x)=25x^2-30x+16#. It appears as if the range, or all the y-values that the graph "covers" starts at about #7# and continues on to #oo#.

We can determine the range algebraically. The vertex of the parabola is the lowest point of the function, so, if we can determine its y-value, we know where the range begins.

So, to figure out the location of the vertex, we can use the vertex formula for a parabola #(-(b)/(2a),f(-(b)/(2a)))#. The #a# and #b# come from the standard form of the parabola #ax^2+bx+c#, so, for #25x^2-30x+16#, #a=25# and #b=-30#.

First, we figure out the x-coordinate of the vertex by plugging in #a# and #b#.
#-(-25)/(2xx30)=25/60=5/12#

Then, to figure out the y-coordinate (this is the one that matters), we find #f(5/12)#, that is, plug #5/12# into the original equation.

We get:
#f(5/12)=25(5/12)^2-30(5/12)+16#
#=25(25/144)-30(60/144)+2304/144#
#=625/144-1800/144+2304/144#
#=color(blue)(1129/144#

Therefore, the coordinate of the vertex is #(5/12,1129/144)#.
This means that the function's lowest y-value is #1129/144#, so the range can be written as #y>=1129/144,{y|y>=1129/144}# or #[1129/144,oo)#.