Video: Graph Transformations of Exponential Functions

William has graphed three exponential functions all in the form 𝑦 = π‘Žπ‘’^π‘₯. The red graph represents the equation 𝑦 = 𝑒^π‘₯. By considering properties of graphical transformations, determine the equation of the blue graph. [A] 𝑦 = 2𝑒^π‘₯ [B] 𝑦 = 3𝑒^π‘₯ [C] 𝑦 = (1/ 2)𝑒^π‘₯ [D] 𝑦 = 𝑒^(π‘₯ βˆ’ 1) [E] 𝑦 = 𝑒^(π‘₯ βˆ’ 0.5)

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Video Transcript

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

We want to consider the properties of transformations. Our starting point is the red graph, 𝑦 equals 𝑒 to the π‘₯ 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 π‘₯ 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 π‘₯, 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 π‘₯ 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 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 𝑦 equals one-half times 𝑒 to the π‘₯ minus half? b 𝑦 equals one-fourth times 𝑒 to the π‘₯ power? c 𝑦 equals one-half times 𝑒 to the π‘₯ power? d 𝑦 equals seven-tenths times 𝑒 to the π‘₯ minus half power? Or e 𝑦 equals 𝑒 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 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 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 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 π‘₯ power. The red graph was 𝑦 equals 𝑒 to the π‘₯ power. The blue graph is 𝑦 equals three times 𝑒 to the π‘₯ power; it’s a stretch. And the green graph is 𝑦 equals one-half times 𝑒 to the π‘₯ power; it’s a compression.

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