Question Video: Determining the Type of Transformation Used on the Graph of an Exponential Function | Nagwa Question Video: Determining the Type of Transformation Used on the Graph of an Exponential Function | Nagwa

Question Video: Determining the Type of Transformation Used on the Graph of an Exponential Function Mathematics • Second Year of Secondary School

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If the function 𝑓(π‘₯) = 2^π‘₯, and the function 𝑓(π‘Žπ‘₯) = 5^π‘₯, find the value of π‘Ž, and state what type of transformation of the graph 𝑦 = 𝑓(π‘₯) this represents.

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Video Transcript

If the function 𝑓 of π‘₯ is equal to two raised to the power π‘₯ and the function 𝑓 of π‘Žπ‘₯ equals five raised to the power π‘₯, find the value of π‘Ž, and state what type of transformation of the graph 𝑦 equals 𝑓 of π‘₯ this represents.

We’re given two functions 𝑓 of π‘₯ equals two raised to the power π‘₯ and 𝑓 of π‘Žπ‘₯ equals five raised to the power π‘₯. And we want to know what value of π‘Ž do we multiply π‘₯ by so that two raised to the power π‘Žπ‘₯ is equal to five raised to the power π‘₯.

Now these are both exponential functions, and we can use the laws of logarithms to simplify this type of function. In particular, we use the power law for logarithms, which tells us that the log to the base 𝑏 of π‘š raised to the power 𝑛 is equal to 𝑛 times the log to the base 𝑏 of π‘š. That is, we bring the power or exponent down to the front and multiply by it.

So if we take the logarithm on both sides of our equation, say, to the base 10, where by convention we don’t write in the base, we have log of two raised to the power π‘Žπ‘₯ is equal to the log of five raised to the power π‘₯. Now using the power law for logs, we can bring both our exponents down to the front and multiply by them. So we have π‘Žπ‘₯ times log two is equal to π‘₯ times log five. And now, provided π‘₯ is nonzero, we can divide through by π‘₯ log two. Dividing top and bottom on the right by π‘₯ then leaves us with π‘Ž equal to log five over log two. So we’ve found our value for π‘Ž. That’s log five over log two.

Making a note of this and making some space, we see that we’re also asked to state what type of transformation of the graph 𝑦 equals 𝑓 of π‘₯ this represents. To answer this, we recall that for a constant 𝑐 greater than zero, replacing π‘₯ with 𝑐 times π‘₯ corresponds to a horizontal scaling of the graph 𝑦 equals 𝑓 of π‘₯. What this means is that π‘₯-values of the graph of 𝑦 equals 𝑓 of π‘₯ are divided by the constant 𝑐. In our case then, with π‘Ž equal to log five over log two corresponding to the constant 𝑐, the transformation of the graph 𝑦 equals 𝑓 of π‘₯ is a horizontal scaling.

It’s worth pointing out that a vertical scaling, as opposed to a horizontal scaling, of 𝑦 equals 𝑓 of π‘₯ occurs when each of the 𝑦-values is multiplied by a constant 𝑐. It’s also worth noting that in the case of a horizontal scaling of the graph 𝑦 equals 𝑓 of π‘₯, a value of the constant 𝑐 greater than one represents horizontal shrinkage or compression of the graph, whereas if 𝑐 is between zero and one, the transformation is a horizontal stretch of the graph 𝑦 equals 𝑓 of π‘₯. Now in our case, with the constant log five over log two, this is greater than one. So this corresponds more precisely to a horizontal shrinkage or compression of the the graph 𝑦 equals 𝑓 of π‘₯.

Finally, it’s important to understand the distinction between algebraic scaling of the variable in the expression, that’s 𝑓 of 𝑐π‘₯, and what happens to the graph under this transformation. In fact, the graph 𝑦 equals 𝑓 of π‘₯ is scaled by a factor of one over the constant 𝑐. And to reiterate, if 𝑓 of π‘₯ is two to the power π‘₯ and 𝑓 of π‘Žπ‘₯ is five to the power π‘₯, then π‘Ž is equal to log five over log two. And this represents a horizontal scaling of the graph 𝑦 equals 𝑓 of π‘₯.

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