Use shifts and scalings to transform the graph of

Question

ƒ (x) = x^{2}

into the graph of g. Use a graphing utility to check your work.

$g (x) = 6 ƒ (\frac{x - 2}{3}) + 1$

Accepted Answer

Welcome back, everyone. Apply the transformations on the graph of P of X equals X squared into the graph of H of X equals five P of X minus four divided by six plus three. Check your work with the help of a graphing utility. We're given four answer choices. They discuss horizontal stretch or compress a shift to the left or to the right, a vertical stretch and a shift up, right. So essentially we want to find the correct combination for this transformation. Let's look at the given function and let's begin with the fact that we can rewrite each of X as follows. We're going to take five P off and we're going to take our denominator. We're simply going to rewrite it as 1/6. And then in pre C we're going to have X minus four, right. So we have a coefficient in front of X minus four and this coefficient is simple 1/6. And then we're adding three. And this helps us understand that first of all, we have a stretch right? There's 1/6 because this specific coefficient is a fractional coefficient where we have one in the numerator and six in the denominator that denominator specifically tells us what type of horizontal stretch we have. And specifically, since it's, it is a fraction, right, the number in front of X is lower than one, we're going to observe a stretch, it is not going to be a compress. So essentially, the first thing that we want to understand is that we are observing a horizontal stretch. This is because the leading coefficient is less than one and the factor would be six. Because whenever we have a numerator of one, the factor itself is going to be the value of the denominator. So we have the correct horizontal shift. Now, let's analyze X minus four itself. That would be number two. And we have to recall that whenever we are subtracting a number from X, this essentially represents a shift to the right. So we have a shift to the right bye or units. If we're adding a number, for example, if we had X plus four, then we would shift to the left. If we're subtracting a number, this represents a shift to the right. So we have a shift to the right by four units. Now, number three, we also have five in front of P, right. And since five is multiplied by the function, the stretch would be a vertical stretch by a factor of five. So we also have a vertical stretch. Vertical stretch is essentially the number in front of the function. Since we have a number five in front of P, this essentially means that we are stretching a function vertically by a factor of five. And now number four, we have the final number which is plus three. This is a vertical shift. If it's positive, if it's going up, if it's negative, it's going down. All of our answer choices essentially state that we are performing a shift up, right. So our vertical shift up corresponds to we units. And now, if we look at all of these, we notice that we have a horizontal stretch by a factor of six right shift by four units, vertical stretch by a factor of five and shift up by three units. Well, this essentially corresponds to the answer choice D. So first of all, let's label D as the correct answer. And now let's focus on the graph. If we use a graphing utility, we can essentially notice that we first of all sketch the function itself. Now the red curve represents X squared and the green one represents the given transformation H of X. So we can essentially label them. We have P of X and we have each of X right. They are labeled. And we can see that the green curve has the correct vertical shift right. We have a shift by three units. It also has a vertical stretch by a factor of five shift to the right by four units. We can notice that as well. Looking at the vertex and also we can notice a significant horizontal stretch, right, the green curve is much wider. So we are essentially observing a horizontal stretch by a factor of sex and that's it. Thank you for watching.

Use shifts and scalings to transform the graph of $ƒ (x) = x^{2}$ into the graph of g. Use a graphing utility to check your work.
$g (x) = 6 ƒ (\frac{x - 2}{3}) + 1$

Key Concepts

Transformations of Functions

Horizontal and Vertical Shifts

Scaling and Stretching

Watch next