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

Function Transformations

This concept refers to the operations used to change the graph of a function, such as translations (shifting), reflections, and scaling. These operations allow us to derive new functions from a base function by systematically altering its equation, which in turn modifies the graph's position, orientation, and shape.

Vertical Translation

Vertical translations shift the graph of a function up or down without changing its shape. This occurs when a constant is added to or subtracted from the function, transitioning the graph vertically and thus affecting the y-intercepts and overall position in the coordinate plane.

Horizontal Translation

Horizontal translations move the graph left or right. This transformation happens when the input variable is adjusted by a constant value, which shifts the graph along the x-axis. The process affects the x-intercepts and the alignment of the graph relative to the y-axis.

Reflection

Reflection changes the orientation of a graph by flipping it across an axis. When the function is multiplied by -1, it reflects the graph across the x-axis; similarly, other forms of reflection can flip the graph horizontally. Reflections alter the signs of the function's outputs or inputs and thereby change the positions of the intercepts.

Vertical Scaling

Vertical scaling involves stretching or compressing a graph along the y-axis. Multiplying a function by a constant factor increases or decreases the graph’s steepness. A factor greater than 1 stretches the graph vertically, while a factor between 0 and 1 compresses it, affecting the distance between key points like intercepts.

Intercepts Identification

Determining the intercepts involves finding where the graph crosses the axes. The x-intercepts are obtained by setting the function equal to zero, whereas the y-intercept is found by evaluating the function at zero. Understanding intercepts is essential for sketching accurate graphs, especially after performing various transformations.

Transcript

00:02 All right.

00:04 We want to sketch four graphs, and they're all going to be based on the curve.

00:13 Y equals x cube.

00:14 That's the parent function.

00:16 So the cubic function looks something kind of like this, passing through the origin.

00:25 We would have the coordinate 1 -1, coordinate negative 1 -1.

00:29 It's an odd function.

00:31 Reflects through the origin.

00:33 So we're going to use that idea of what that parent function looks like.

00:44 And perform these transformations to graph these functions.

00:48 So if p of x is equal to x cubed minus eight, this is simply the graph of y equals x cubed shifted down eight units.

01:05 So that means that coordinate that's normally at the origin is now gonna be compressing through the coordinate zero negative eight.

01:18 And we have this curve, when something like that.

01:22 So zero negative eight.

01:23 Is our y intercept.

01:27 And our x intercept, if we let the function equal zero, x cubed minus eight equal zero.

01:38 It means if i add eight to each side, x cubed is equal to eight.

01:42 And if i take the cube root of each side, x is equal to two.

01:47 So i have an x intercept at the coordinate to zero.

01:55 The next one we want to look at is q of x, which is going to be negative x cubed, plus 27.

02:08 So there's two transformations going on with this one.

02:11 I have a shift up of 27 units.

02:18 I'm adding 27 to the function.

02:21 It's not inside parentheses with the x, so that's being added to the y values.

02:27 And this negative out in front of the function indicates that we are going to reflect over the x -axis.

02:42 So our curve is going to be upside.

02:46 Down and it's going to be shifted up 27 units.

02:53 So that means our curve be doing something like that.

03:01 And once again, our y intercept is going to be the coordinate 0, 27.

03:09 And we're going to have an x intercept down here.

03:12 If negative x cubed plus 27 were equal to zero, i would add x cubed to each side.

03:21 So 27 equals x cubed.

03:26 I could then take the cube root of each side, and we find out that we have an x intercept at the coordinate 3 .0.

03:41 Part c, r of x is equal to negative x plus 2 cubed.

03:54 So once again, we have a couple of transformations going on here.

04:00 That negative out front once again indicates that we're reflecting over the x -axis...

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