How do you find the domain and range of #y = (x + 7)^2 - 5#?

Question

Nathan Axel · Accepted Answer

#D: (-oo, oo)#
#R: [-5, oo)# You can find quadratics in two ways: #f(x)=ax^2+bx+c# #color(blue)(" Standard Form")# #f(x)=a(x-h)^2+k# #color(blue)(" Vertex Form")# Obviously we'll ignore the #"standard form"# for this problem, but it's important to know both. Since our equation is in #"vertex"# form, we're given the #"vertex"# without having to solve for it: #"Vertex: " (-h, k)# Don't forget that the default vertex is #-h#, don't forget the negative! Let's look back at our original equation: #f(x)=(xcolor(red)(+7))^2color(red)(""-5)# Let's plug in our #h# and #k# values into the #"vertex point:"# #(-h, k)# #((-)+7, -5)# #color(red)((-7, -5)# Notice that a negative and a positive make a negative, hence the #-7# even though it's #+7# in the original equation. Now that we know our #"vertex"#, solving for domain and range is very easy #"Domain: All x-values"# The good thing with this problem is that all quadratics will always have an infinite domain of #" all real numbers "# since the graph goes on infinite horizontally and vertically (upward). So: #color(red)(D: (-oo, oo))# #"Range: All y-values"# With both #"Domain"# and #"Range"# we measure from lowest to highest, and the lowest point on this quadratic is the #y"-coordinate"# of the vertex since the graph opens upward infinitely. So: #color(red)(R: [-5, oo))# It's crucial to keep in mind that any value that is included or "touched" on the domain and/or range graphs needs to be enclosed in brackets. If the value is enclosed in parentheses, it indicates that it rises to that value but does not touch it, similar to an asymptote. Since we obviously cannot touch infinity, we leave those enclosed in parentheses; however, the graph touches -5, so we use brackets on that portion of the graph but not the infinity. The #"Domain"# reads as #"the graph includes all x-values,"# since the quadratic does not end horizontally. The #"Range"# reads as #"The graph starts at "-5" and extends upward infinitely."# If you're still unclear, you can always picture it this way: chart{(x+7)^2-5 [-10, 10, -5, 5]}

Eva Adamson · Answer

undefined To find the domain and range of $ y = (x + 7)^2 - 5 $:

1. Domain: The domain is all real numbers because the function is defined for all values of $ x $.

2. Range: To find the range, consider the vertex of the parabola $ y = (x + 7)^2 - 5 $, which occurs at $ (-7, -5) $. Since the parabola opens upwards, the range is $ y \geq -5 $.