Video Transcript
William has graphed three
exponential functions all in the form 𝑦 equals 𝑎𝑒 to the 𝑥 power. The red graph represents the
equation 𝑦 equals 𝑒 to the 𝑥 power. By considering properties of
graphical transformations, determine the equation of the blue line. a) 𝑦 equals two
times 𝑒 to the 𝑥 power. b) 𝑦 equals three times 𝑒 to the 𝑥 power. c) 𝑦 equals
one-half times 𝑒 to the 𝑥 power. d) 𝑦 equals to the 𝑥 minus one power. Or e) 𝑦 equals 𝑒 to the 𝑥 minus
one-half power.
We want to consider the properties
of transformations. Our starting point is the red
graph, 𝑦 equals 𝑒 to the 𝑥 power. And we’re considering the way that
the red graph has been transformed to be the blue graph. It looks like the red graph has
been stretched to create the blue graph in a vertical direction.
We can stretch or compress a
function in the 𝑦-direction by multiplying the whole function by a constant. If that constant is greater than
one, it’s a stretch. If the constant is between zero and
one, if the constant is a fraction, it’s a compression. We visually see that the blue graph
is a stretch, and that means we won’t be multiplying by one-half. It also means we’re not changing
the exponents.
We need to determine if we’ve
multiplied 𝑒 to the 𝑥 power by two or three. Notice that the red graph has a
𝑦-intercept at the point zero, one, has an intercept at the point zero, three. If we substitute zero in for 𝑥, 𝑒
to the zero power equals one. Two times one and three times
one. 𝑦 equals two or 𝑦 equals
three. Remember, we were solving for the
place where 𝑥 equals zero and we need 𝑦 equal to three. By multiplying the whole equation
by three, we’ve performed a stretch. The blue graph is the function 𝑦
equals three times 𝑒 to the 𝑥 power.
We now want to consider the
properties of graphical transformations to determine the equation for the green
graph. Would the function be a 𝑦 equals
one-half times 𝑒 to the 𝑥 minus half? b 𝑦 equals one-fourth times 𝑒 to the 𝑥
power? c 𝑦 equals one-half times 𝑒 to the 𝑥 power? d 𝑦 equals seven-tenths times
𝑒 to the 𝑥 minus half power? Or e 𝑦 equals 𝑒 to the 𝑥 power
minus one-half?
The same thing that was true for
part a is true again. When we stretch or compress a
function in the 𝑦-direction, we multiply the function by a constant. To go from the red graph to the
green graph is a compression in the 𝑦-direction. And that means we’ll be multiplying
whatever the red equation was by a fraction. However, it does not change the
exponents. And that means any answer choice
that’s modifying the exponent can’t be right.
We’ll use the same strategy and
consider the 𝑥-intercept. We know that the 𝑥-intercept of
the red graph is zero, one and the 𝑥-intercept of the green graph is zero,
one-half. And that means we’ve multiplied the
equation by one-half. The new graph, or the graph of the
green function, is 𝑦 equals one-half times 𝑒 to the 𝑥 power. The red graph was 𝑦 equals 𝑒 to
the 𝑥 power. The blue graph is 𝑦 equals three
times 𝑒 to the 𝑥 power; it’s a stretch. And the green graph is 𝑦 equals
one-half times 𝑒 to the 𝑥 power; it’s a compression.
