Sketch the graph of each function by transforming the graph of an appropriate function of the fo

Sketch the graph of each function by transforming the graph...

Key Concepts

Intercepts

Intercepts are points where the graph of a function crosses the axes. The x-intercepts are found by setting the function equal to zero and solving for x, while the y-intercept is obtained by evaluating the function at x = 0. Identifying intercepts is an essential part of graphing functions because they provide fixed points that help in anchoring the position and shape of the graph relative to the coordinate axes.

Reflection

Reflection is a transformation that flips the graph of a function over a specific axis. Reflecting across the x-axis, for instance, changes the sign of the function’s output, turning f(x) into -f(x). This operation reverses the graph vertically, altering the positions of intercepts and the overall orientation of the original graph. Such reflections are key when dealing with functions that may have a mirror image relative to the x-axis.

Vertical Dilation (Scaling)

Vertical dilation or scaling involves stretching or compressing the graph of a function in the vertical direction. Multiplying the function by a constant factor greater than 1 stretches the graph vertically, making the peaks and valleys more pronounced, while a factor between 0 and 1 compresses it, reducing the vertical distances. This transformation affects the steepness of the graph and the spacing of its features, such as intercepts and extrema.

Horizontal Translation

A horizontal translation shifts the graph of a function left or right without altering its overall shape. This transformation is reflected in functions when x is replaced by (x - h) or (x + h), where h is a constant. The graph shifts to the right by h units when written as (x - h) and to the left when written as (x + h), thereby moving the location of key features such as intercepts without changing the graph's orientation.

Graph Transformations

Graph transformations involve modifying the basic shape of a function’s graph through operations such as translation, scaling, and reflection. These modifications are applied to a standard or base function to obtain a new function, and they allow a systematic way of understanding how the graph’s features change. In the context of polynomial functions, such as those of the form y = x^n, these transformations help in sketching graphs by shifting, stretching, compressing, or reflecting the original graph.

Vertical Translation

Vertical translation moves the graph up or down without changing its shape. This is achieved by adding or subtracting a constant outside the function. When a function f(x) is transformed to f(x) + k, the whole graph is shifted upwards by k units if k is positive, and downwards if k is negative. This type of transformation impacts the vertical location of key points including the y-intercept.

Transcript

00:06 All right.

00:09 So we have four curves that we have to sketch, and they're based on the graph of y equals x to the fifth power.

00:19 The graph of y equals x to the fifth power looks like the graph of the cubic function.

00:27 So in other words, it comes like this.

00:29 It flattens out a little bit here in the middle and goes back up.

00:35 So that's the parent function that we're going to be basing all of our graphs on.

00:40 And we're going to graph using the rules of transformations.

00:46 So if p of x is equal to x plus three to the fifth power, then this is going to be the graph of x to the fifth power.

00:58 Now this plus three is being added to the x.

01:00 It's inside the function.

01:03 This right here is a shift to the left of three units.

01:13 So when we sketch this, the coordinate that would ordinarily be at the origin, instead is going to be over here at negative 3 -0, and we're just going to simply have a curve that looks just like the original does that's passing through that coordinate.

01:35 So we have an x intercept at negative 3 -0.

01:41 That will be our x intercept, and our y intercept is going to be right here where x is equal to zero.

01:48 Well, if we plug zero in for x, we're going to get 3 to the 5th power, and this right here is going to be the coordinate 0, 243.

02:00 So that's our y intercept, and our x intercept is at negative 3 .0.

02:05 The second function we want to look at then, q of x, is going to be 2 times x plus 3 to the 5th power, minus 64.

02:23 So there's three transformations going on with this one.

02:27 We have a shift to the left of three units, just like the first one.

02:32 So we're going to the left three.

02:36 This right here, outside the function, it's not being raised to the fifth power.

02:39 So that's a change to the y.

02:41 So this is a shift down 64 units.

02:47 And here, we're multiplying the function by two.

02:51 That's going to be a vertical stretch by a factor of two.

02:59 In other words, we're doubling all of the y values.

03:02 So what is this going to end up looking like? well, this coordinate, once again, that's normally at the origin, is getting shifted to the left three units, but it's also getting shifted down 64 units.

03:26 So we're going to have this curve that's coming up and doing something like this.

03:36 So what do we have going on for intercepts? well, y intercept is going to be, actually our y intercept should be above the x axis, shouldn't it? it's going to be a positive value.

03:57 It's going to be getting large quickly, because if we let x equal zero, then q of zero is going to be two times zero plus three to the fifth power minus 64.

04:21 That's going to be two times 243 minus 64.

04:27 That's going to give us a y intercept of 422...

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