More Advanced Parent Functions #1 (Square Root, Cubic, Cube Root & Rational)
Description/Explanation/Highlights
Video Description
This video focuses on the characteristics of the square root, cubic, cube root and rational parent functions.
Steps and Key Points to Remember
To recognize and describe the characteristics of the square root, cubic, cube root, and rational parent functions, follow these steps:
- Square root, cubic, cube root, and rational functions are all function families characterized by certain characteristics that start with the simplest form of the function, its parent function.
- Square root functions are recognized symbolically by the \(\sqrt{\;\;}\) symbol. Its parent function is \(y=\sqrt{x}\).
- In order to use the parent function and the square root function family, we must choose numbers carefully!
- When choosing values for x in the parent function, we cannot use negative numbers as square roots of negative numbers do not result in real answers.
- Since we will not be using negative x-values, the resulting parent graph will not have values left of the y-axis.
- We also want to choose x-values carefully in our graph so that we get numbers that we can find and graph the square roots of more easily such as: 0, 1, 4, 9
- Also, recall that square roots have two answers; a positive and a negative version. We will limit the y-values to the positive results to keep it the square root function.
- So the coordinates we use to graph the square root parent are its starting point at (0, 0) and (1, 1) (4, 2) & (9, 3)
- Notice how graphing these points gives us half of a rather flat parabola turned on its side.
- Note also that had we graphed the negative versions of the square roots, we would have graphed the bottom half of the parabola and the graph would no longer have been a function as it would have violated the vertical line test.
- The parent function of the cube function is \(y=x^3\)
- To graph the parent, we choose some simple values of x to plug in to find the coordinates. For example: (-2, -8) (-1, -1) (0, 0) (1, 1) (2, 8)
- When these coordinates are graphed, the graph reveals half of a parabola that opens downward to the left of the y-axis and below the x-axis and half of a parabola that opens upward to the right of the y-axis and above the x-axis.
- (0, 0) marks the point where the parabola halves change directions.
- \(y=\sqrt[3]{x}\) is the parent function of the cube root family.
- The graph of the cube root parent looks like the square root parent to the right of the y-axis except that it is flatter. However, the cube root parent has values to the left of the y-axis that create half of a sideways parabola below the x-axis that is a 180 degree rotation around the origin from the right side.
- As with the square root parent, choosing x-values that are easy to find the cube root of will help in creating the graph. The following coordinates are recommended: (-8, -2) (-1, -1) (0, 0) (1, 1) (8, 2)
- \(y=\frac{1}{x}\) is the parent function for the rational function family.
- Rational functions are recognized by fractions with a variable (x) in the denominator.
- Graphing the rational parent creates asymptotes, which are places the graph approaches but never reaches, at the x- and y-axis.
- The entire graph of the rational parent is in quadrant I and III.
- The graph approaches the x-axis from below and the y-axis from the left in quadrant III and the x-axis from above and the y-axis from the right in quadrant I.
- Points (-1, -1) and (1, 1) mark the point that the graph changes which axis it is approaching in each quadrant.
- The graph never reaches the x- or y-axis and the function is undefined at x = 0.
- Caution: The graph of \(y=-\frac{1}{x}\) is a reflection over the x-axis of the parent function but is graphed in quadrants II & IV. It is not the parent function!
Here are some key points to keep in mind when identifying characteristics of parent functions and their graph.
- The parent of the square root function is: \(y=\sqrt{x}\)
- Negative values for x are undefined in the square root parent.
- For the square root parent function to remain a function, use only the positive square roots of numbers.
- The graph starts at (0, 0) and makes half of a sideways parabola in quadrant I.
X
0
1
4
9
Y
0
1
2
3
The table of values to the left is found by substituting the x-values into the \(y=\sqrt{x}\) parent function. Graphing these coordinates will give a good representation of the square root parent graph and its shape.
- The parent of the cubic function is \(y=x^3\)
- The cubic parent makes half of a downward opening parabola on the left of the y-axis and half of an upward opening parabola on the right side of the y-axis.
- (0, 0) marks the point where the graph changes from opening downward to opening upward.
- The top half of the graph is in quadrant I and when rotated 180 degrees around the origin will make the bottom half of the graph in quadrant III.
X
-2
-1
0
1
2
Y
-8
-1
0
1
8
The table of values to the left is found by substituting the x-values into the \(y=x^3\) parent function. Graphing these coordinates will give a good representation of the cubic parent graph and its shape.
- The parent of the cube root function is \(y=\sqrt[3]{x}\)
- The right side of the cube root function looks like the square root function only flatter.
- Unlike the square root function, the cube root function is defined on x-values to the left of (0, 0)
- The left side forms a sideways half-parabola in quadrant III that is a 180 degree rotation of the right side from quadrant I.
X
-8
-1
0
1
8
Y
-2
-1
0
1
2
The table of values to the left is found by substituting the x-values into the \(y=\sqrt[3]{x}\) parent function. Graphing these coordinates will give a good representation of the cube root parent graph and its shape.
- The parent of the rational function is \(y=\frac{1}{x}\)
- The entire graph will appear in quadrant I & III.
- The graph approaches the x-axis from below and the y-axis from the left in quadrant III.
- The graph approaches the x-axis from above and the y-axis from the right in quadrant I.
- The x- and y-axis form asymptotes that the graph approaches but never reaches in the graph of the parent function.
- (-1, -1) & (1, 1) are the points that mark the place where the graph changes from approaching the x-axis to approaching the y-axis in quadrants I & III.
- The value of y is undefined at x = 0.
- Anything graphed in quadrant II or IV does not represent the graph of the parent function.
X
-4
-1
-1/2
-1/4
0
1/4
1/2
1
4
Y
-1/4
-1
-2
-4
undef.
4
2
1
1/4
The table of values to the left is found by substituting the x-values into the \(y=\frac{1}{x}\) parent function. Graphing these coordinates will give a good representation of the rational parent graph and its shape.
Video Highlights
- 00:00 Introduction
- 00:38 Characteristics of the \(y=\sqrt{x}\) square root parent function
- 04:21 Characteristics of the \(y=x^3\) cubic parent function
- 07:00 Characteristics of the \(y=\sqrt[3]{x}\) cube root parent function
- 09:20 Characteristics of the \(y=\frac{1}{x} \) rational parent function
- 14:00 Conclusion
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