# Question Video: Graph Transformations of Exponential Functions Mathematics

William has graphed three exponential functions all in the form π¦ = ππ^π₯. The red graph represents the equation π¦ = π^π₯. By considering properties of graphical transformations, determine the equation of the blue graph. [A] π¦ = 2π^π₯ [B] π¦ = 3π^π₯ [C] π¦ = (1/ 2)π^π₯ [D] π¦ = π^(π₯ β 1) [E] π¦ = π^(π₯ β 0.5)

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### Video Transcript

William has graphed three exponential functions all in the form π¦ equals ππ to the π₯ power. The red graph represents the equation π¦ equals π to the π₯ power. By considering properties of graphical transformations, determine the equation of the blue line. a) π¦ equals two times π to the π₯ power. b) π¦ equals three times π to the π₯ power. c) π¦ equals one-half times π to the π₯ power. d) π¦ equals to the π₯ minus one power. Or e) π¦ equals π to the π₯ minus one-half power.

We want to consider the properties of transformations. Our starting point is the red graph, π¦ equals π to the π₯ power. And weβ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 π¦-direction by multiplying the whole function by a constant. If that constant is greater than one, itβs a stretch. If the constant is between zero and one, if the constant is a fraction, itβs a compression. We visually see that the blue graph is a stretch, and that means we wonβt be multiplying by one-half. It also means weβre not changing the exponents.

We need to determine if weβve multiplied π to the π₯ power by two or three. Notice that the red graph has a π¦-intercept at the point zero, one, has an intercept at the point zero, three. If we substitute zero in for π₯, π to the zero power equals one. Two times one and three times one. π¦ equals two or π¦ equals three. Remember, we were solving for the place where π₯ equals zero and we need π¦ equal to three. By multiplying the whole equation by three, weβve performed a stretch. The blue graph is the function π¦ equals three times π to the π₯ power.

We now want to consider the properties of graphical transformations to determine the equation for the green graph. Would the function be a π¦ equals one-half times π to the π₯ minus half? b π¦ equals one-fourth times π to the π₯ power? c π¦ equals one-half times π to the π₯ power? d π¦ equals seven-tenths times π to the π₯ minus half power? Or e π¦ equals π to the π₯ 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 π¦-direction, we multiply the function by a constant. To go from the red graph to the green graph is a compression in the π¦-direction. And that means weβll be multiplying whatever the red equation was by a fraction. However, it does not change the exponents. And that means any answer choice thatβs modifying the exponent canβt be right.

Weβll use the same strategy and consider the π₯-intercept. We know that the π₯-intercept of the red graph is zero, one and the π₯-intercept of the green graph is zero, one-half. And that means weβve multiplied the equation by one-half. The new graph, or the graph of the green function, is π¦ equals one-half times π to the π₯ power. The red graph was π¦ equals π to the π₯ power. The blue graph is π¦ equals three times π to the π₯ power; itβs a stretch. And the green graph is π¦ equals one-half times π to the π₯ power; itβs a compression.