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 the points where the graph crosses the axes. The x-intercepts are found by solving for x when y is zero, and the y-intercept is determined by evaluating the function at x = 0. Understanding intercepts helps in sketching the graph accurately and in understanding the function’s behavior at key points.

Vertical Translations

Vertical translations move the graph of a function up or down. This is accomplished by adding or subtracting a constant from the entire function. Adding a constant k to the function will shift the graph upward by k units, and subtracting k will shift it downward by k units.

Vertical Scaling and Reflection

Vertical scaling involves stretching or compressing the graph of a function by multiplying it by a constant factor. When the constant is greater than 1, the graph stretches vertically; when between 0 and 1, it compresses. If the constant is negative, it also reflects the graph across the x-axis, which inverts the orientation of all y-values.

Graph Transformations

Graph transformations involve altering the basic graph of a function through various operations such as translations (shifts), reflections, and scalings. These transformations change the position, shape, and orientation of the graph while retaining its overall structure, and they are key to understanding how more complex functions are derived from simpler ones.

Even-Powered Functions

Even-powered functions, such as y = x?, typically have a symmetric, U-shaped graph that is similar to the parabola of a quadratic function but with a steeper rise on the ends. They are characterized by having a minimum (or maximum if reflected) at the origin and are symmetric about the y-axis.

Horizontal Translations

Horizontal translations involve shifting the graph of a function left or right. This is achieved by adding or subtracting a constant inside the function's argument. For instance, replacing x with (x + c) will shift the graph to the left by c units, while replacing x with (x - c) will shift it to the right by c units.

Transcript

00:04 So we have four curves that we want to sketch, but they're all based on the parent function where y is equal to x to the fourth power.

00:20 And the graph of y equals x to the fourth power looks like the graph of y equals x squared just flattens out a little bit more here in the middle.

00:31 So that would be the graph of y equals x to the fourth.

00:33 So all of these other functions are transformations of that function.

00:43 So the first graph that we're going to look at, the first curve we're going to sketch, is p of x, which is equal to x to the fourth power minus 16.

00:55 Now, we have a lot of transformation rules.

00:59 One rule states that if we have y equals f of x minus k, we're subtracting k from the function values.

01:11 Then that's a shift down of k units.

01:15 So in this particular case, we're shifting all of our coordinates down 16 units.

01:24 So what that means then is the graph of this function is going to look just like the graph of y equals x to the fourth power, except this coordinate that's right here.

01:43 Right, the vertex, which is at the coordinate zero zero, now is going to get shifted down to the coordinate 0 -1916.

01:58 So our curve will be doing something like that.

02:06 And as far as our x intercepts go, well, x intercepts are easy enough to find.

02:16 If we take the function and let it equals zero, we can actually factor and solve this thing.

02:24 This is the difference of two squares.

02:26 I can factor to this the x squared plus four times x squared minus four now x squared minus four can't be factored again but the x or x squared plus four cannot be factored again but the x squared minus four can and that can be factored to x plus two times x minus two so if x plus two is equal to zero then we have a root at negative two and if x minus two is equal to zero then our other root is that positive two so now we have x intercept at 2 .0, negative 2 .0, and a y intercept at 0 negative 16.

03:13 Part b, q of x is x plus 2 to the 4th power.

03:24 Now, when we have a function where we are adding or subtracting something inside the function just to x, that's a horizontal transformation.

03:38 And horizontal transformations always work backwards from the way you think.

03:42 So this right here, when i say x plus 2 like this, this indicates that we are shifting all of our coordinates to the left two units.

03:55 So this curve, once again, is going to look just like the curve of y equals x to the fourth, except that coordinate that once again is usually at the origin is now going to get shifted to the left two units.

04:17 And it's going to look something like this, where the x intercept now, is that the coordinate negative to 0.

04:29 And we're going to have a y intercept.

04:33 The y intercept always occurs where x is equal to zero.

04:37 So if i were to evaluate q of zero, i'm going to have zero plus two to the fourth power.

04:44 That'll be two to the fourth power.

04:46 That's going to make that the coordinate 0 16 for the y intercept.

04:56 Part c has us combine parts a and b, where we're going to have x plus 2 to the fourth power minus 16.

05:09 So now we're going to the left 2 and down 16...