How do you sketch the graph of #y=2(x+4)^2-3# and describe the transformation?

1 Answer
Jan 9, 2018

See Explanation. Transformations are vertical stretch by a factor of 2, horizontal shift 4 units left, vertical shift 3 units down.

Explanation:

This is a quadratic in vertex form.
#y=a(x-h)^2+k#

#a# is the vertical stretch. If it is big, there is a lot of stretch. If it it less than 1, there is compression.

#h# is the horizontal shift. Notice the "#-#" in the equation.
This means a #h-value# of #+4# as seen in the question is really a #-4#, shifting the graph left.

# k# is the vertical shift. It moves the graph up/down. Positive #k# moves the graph up while negative #k# move the graph down. Simple.

Now the graphs. I do one transformation each time to show the steps.

#y=x^2#
graph{x^2 [-4.75, 5.25, -0.98, 4.02]}

#y=2x^2#
graph{2x^2 [-4.75, 5.25, -0.98, 4.02]}

#y=2(x+4)^2#
graph{2(x+4)^2 [-8.25, 1.75, -0.9, 4.1]}

#y=2(x+4)^2-3#
graph{2(x+4)^2-3 [-8.04, 1.96, -3.22, 1.78]}