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| 14 Theory slides |
| 19 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
The graph of the parent function y=x and the graph of the radical function y=ax−h+k are drawn on the same coordinate plane.
Find the values of a, h, and k.The graph of the function y=x−h+k is shown in the coordinate plane. By changing the values of h and k, observe how the graph is horizontally and vertically translated.
Graph:
Graph:
Graph:
The graphs of the rational function y=x and a vertical or horizontal translation are shown in the coordinate plane.
The graph of the radical function y=acx is shown in the coordinate plane. By changing the values of a and c, observe how the graph is vertically and horizontally stretched and shrunk.
Suppose that a function is horizontally or vertically stretched/shrunk, and that the graphs of the transformed and the original function are both drawn on the same coordinate plane. Then, the values of a or c can be found by following these procedures.
Finding a | Select two points with the same x-coordinate, one point P on the parent function and the other point Q on the transformed function. The value of a is the quotient of the y-coordinate of Q and the y-coordinate of P. |
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Finding c | Select two points with the same y-coordinate, one point P on the parent function and the other point Q on the transformed function. The value of c is the quotient of the x-coordinate of P and the x-coordinate of Q. |
After mastering vertical and horizontal translations of radical functions, Ignacio is having a hard time understanding vertical and horizontal stretches and shrinks of this type of function. He asked for some help from his very good friend Jordan.
In order to help him, Jordan asks Ignacio to consider the following function.Graph:
Graph:
The graph of the parent function y=x is shown in the coordinate plane. The graph of a horizontal or vertical stretch or shrink is also shown.
Ignacio is now feeling pretty confident about transformations of radical functions again. Now he turns his attention to reflections.
Ignacio considers the following radical function.Graph:
Graph:
Ignacio confidently states that he can now solve any exercise about transformations of radical functions. Jordan skeptically challenges her friend to solve an exercise that combines transformations.
Help Ignacio solve Jordan's challenge!Equation: y=-x+5−6
Graph:
Start by multiplying the output by 2 to find the function that represents the vertical stretch. Then, multiply the input by -1 to reflect the graph in the y-axis. Finally, subtract 3 units from the input.
To help Ignacio, the transformations will be applied one at a time.
With the topics learned in this lesson, the challenge presented at the beginning can be solved. The graphs of the radical function y=ax−h+k and its corresponding parent function y=x are given.
Find the values of a, h, and k.Start by considering a vertical stretch. Then, consider vertical and horizontal translations.
Disregard translations for a moment. In the graph of y=x it can be understood as, after moving 1 unit to the right of the starting point (0,0), the graph increases vertically 1 unit. Conversely, in the graph of y=ax−h+k, after moving 1 unit to the right of the starting point (-3,0), the graph increased vertically 2 units.
Let's start by drawing the graph of the given radical function. To do this, we will make a table of values. Since the radicand of a square root must be non-negative, we will only use non-negative values for the variable x.
x | y=sqrt(x)-1 | y |
---|---|---|
0 | sqrt(0)-1 | - 1 |
1 | sqrt(1)-1 | 0 |
2 | sqrt(2)-1 | ≈ 0.4 |
3 | sqrt(3)-1 | ≈ 0.7 |
4 | sqrt(4)-1 | 1 |
5 | sqrt(5)-1 | ≈ 1.2 |
10 | sqrt(10)-1 | ≈ 2.2 |
We can now plot the points obtained in the table and connect them with a smooth curve.
Finally, let's translate this function 2 units up.
This graph corresponds to choice B.
We want to obtain a vertical translation 3 units down of the graph of the given radical function. To do this, we have to subtract 3 from the output of the function rule. Notice that sqrt(x)-1 represents the outputs of the function. cc Function & Translation3 Units Down y=sqrt(x)-1 & y=(sqrt(x)-1) -3 Finally, let's simplify the right-hand side of this equation.
Let's start by drawing the graph of the given radical function. To do this, we will make a table of values. Since the radicand of a square root must be non-negative, we will only use non-negative values for the variable x.
x | y=sqrt(x)-1 | y |
---|---|---|
0 | sqrt(0)-1 | - 1 |
1 | sqrt(1)-1 | 0 |
2 | sqrt(2)-1 | ≈ 0.4 |
3 | sqrt(3)-1 | ≈ 0.7 |
4 | sqrt(4)-1 | 1 |
5 | sqrt(5)-1 | ≈ 1.2 |
10 | sqrt(10)-1 | ≈ 2.2 |
We can now plot the points obtained in the table and connect them with a smooth curve.
Finally, let's translate this graph 3 units to the right.
This graph corresponds to choice A.
We want to obtain a horizontal translation 5 units to the left of the graph of the given radical function. To do this, let's first recall how to perform vertical and horizontal translations.
Translations of f(x) | |
---|---|
Vertical Translations | Translation up k units, k>0 y=f(x)+ k |
Translation down k units, k>0 y=f(x)- k | |
Horizontal Translations | Translation right h units, h>0 y=f(x- h) |
Translation left h units, h>0 y=f(x+ h) |
This review lets us know that we need to add 5 to the input of the function rule. Notice that the input is represented with x. cc Function & Translation5 Units to the Left y=sqrt(x)-1 & y=sqrt(x + 5)-1
Let's start by drawing the graph of the given radical function. To do this, we will make a table of values. Since the radicand of a square root must be non-negative, we will only use non-negative values for the variable x.
x | y=sqrt(x)-1 | y |
---|---|---|
0 | sqrt(0)-1 | - 1 |
1 | sqrt(1)-1 | 0 |
2 | sqrt(2)-1 | ≈ 0.4 |
3 | sqrt(3)-1 | ≈ 0.7 |
4 | sqrt(4)-1 | 1 |
5 | sqrt(5)-1 | ≈ 1.2 |
10 | sqrt(10)-1 | ≈ 2.2 |
We can now plot the points obtained in the table and connect them with a smooth curve.
Finally, let's translate this function 1 unit down and 2 units to the left.
This graph corresponds to choice D.
We want to obtain a translation 3 units to the right and 2 units down of the graph of the given radical function. To do this, we have to subtract 3 from the input and subtract 2 from the output of the function rule. Notice that x represents the input and sqrt(x)-1 represents the outputs of the given function. cc Function & Translation3 Units to the & Right and2 Units Down y=sqrt(x)-1 & y=sqrt(x- 3)-1- 2 Let's finally simplify the right-hand side of the obtained equation.
Let's start by drawing the graph of the given radical function. To do this, we will make a table of values. Since the radicand of a square root must be non-negative, we will only use values greater than or equal to 1 for the variable x.
x | sqrt(x-1)+2 | y |
---|---|---|
1 | sqrt(1-1)+2 | 2 |
2 | sqrt(2-1)+2 | 3 |
5 | sqrt(5-1)+2 | 4 |
10 | sqrt(10-1)+2 | 5 |
We can now plot the points obtained in the table and connect them with a smooth curve.
Finally, let's vertically stretch this graph by a factor of 2.
This graph matches choice A.
We want to obtain a vertical stretch by a factor of 5 of the graph of the given radical function. To do this, we need to multiply by 5 the output of the function rule. cc Function & Vertical Stretch & by a Factor of5 y=sqrt(2x+1)-3 & y= 5(sqrt(2x+1)-3) Let's now simplify the right-hand side of the obtained equation. To do so, we will distribute the 5.
We want to obtain a horizontal shrink by a factor of 3 of the graph of the given radical function. To do this, we need to multiply by 3 the input of the function. Notice that the input of this function is represented by x. cc Function & Horizontal Shrink & by a Factor of3 y=sqrt(2x+1)-3 & y=sqrt(2( 3x)+1)-3 Let's now simplify the right-hand side of the obtained equation.
Let's start by drawing the graph of the given radical function. To do this, we will make a table of values. Since the radicand of a cube root can be any real number, we can use positive values, negative values, and zero for the variable x.
x | sqrt(1/2x-1)+2 | y |
---|---|---|
- 5 | sqrt(1/2( - 5)-1)+2 | ≈ 0.5 |
- 3 | sqrt(1/2( - 3)-1)+2 | ≈ 0.6 |
- 1 | sqrt(1/2( - 1)-1)+2 | ≈ 0.9 |
0 | sqrt(1/2( 0)-1)+2 | 1 |
1 | sqrt(1/2( 1)-1)+2 | ≈ 1.2 |
3 | sqrt(1/2( 3)-1)+2 | ≈ 2.8 |
5 | sqrt(1/2( 5)-1)+2 | ≈ 3.1 |
We can now plot the points obtained in the table and connect them with a smooth curve.
Finally, let's horizontally shrink this graph by a factor of 4. Graphically, this means that the x-coordinate of each point on the transformed curve will be one quarter the x-coordinate of its corresponding point on the original curve. Let's consider the points (2,2) and (4,3).
Point | Transformed Point |
---|---|
( 2, 2) | (1/4( 2), 2) ⇕ (1/2,2) |
( 4, 3) | (1/4( 4), 3) ⇕ (1,3) |
Note that, for each point on the curve, the y-coordinate will remain the same but the x-coordinate will be one quarter its original value.
This graph matches choice B.
Let's start by drawing the graph of the given radical function. To do this, we will make a table of values. Since the radicand of a cube root can be any real number, we can use positive values, negative values, and zero for the variable x.
x | 2sqrt(x+1)-2 | y |
---|---|---|
- 5 | 2sqrt(- 5+1)-2 | ≈ - 5.2 |
- 2 | 2sqrt(- 2+1)-2 | - 4 |
- 1 | 2sqrt(- 1+1)-2 | - 2 |
0 | 2sqrt(0+1)-2 | 0 |
3 | 2sqrt(3+1)-2 | ≈ 1.2 |
5 | 2sqrt(5+1)-2 | ≈ 1.6 |
7 | 2sqrt(7+1)-2 | 2 |
We can now plot the points obtained in the table and connect them with a smooth curve.
Let's finally reflect this graph in the x-axis.
This graph matches choice C.
We want to obtain the equation of a function whose graph is a reflection in the x-axis of the graph of the given radical function. To do so, we need to multiply the function rule by - 1. cc Function & Reflection in thex -axis y=2sqrt(x+1)-2 & y= - (2sqrt(x+1)-2) Now we can distribute - 1 to simplify the right-hand side of the equation.
We want to obtain the equation of a function whose graph is a reflection in the y-axis of the graph of the given radical function. To do so, we need to multiply the input of the function — the variable x — by - 1. cc Function & Reflection in they -axis y=2sqrt(x+1)-2 & y=2sqrt(-x+1)-2
Let's start by drawing the graph of the given radical function. To do this, we will make a table of values. Since the radicand of a cube root can be any real number, we can use positive values, negative values, and zero for the variable x.
x | 2sqrt(x+1)-2 | y |
---|---|---|
- 5 | 2sqrt(- 5+1)-2 | ≈ - 5.2 |
- 2 | 2sqrt(- 2+1)-2 | - 4 |
- 1 | 2sqrt(- 1+1)-2 | - 2 |
0 | 2sqrt(0+1)-2 | 0 |
3 | 2sqrt(3+1)-2 | ≈ 1.2 |
5 | 2sqrt(5+1)-2 | ≈ 1.6 |
7 | 2sqrt(7+1)-2 | 2 |
We can now plot the points obtained in the table and connect them with a smooth curve.
Let's finally reflect this graph in the y-axis.
This graph matches choice D.
Let's start by drawing the graph of the given radical function. To do this, we will make a table of values. Since the radicand of a cube root can be any real number, we can use positive values, negative values, and zero for the variable x.
x | 2sqrt(x+1)-2 | y |
---|---|---|
- 5 | 2sqrt(- 5+1)-2 | ≈ - 5.2 |
- 2 | 2sqrt(- 2+1)-2 | - 4 |
- 1 | 2sqrt(- 1+1)-2 | - 2 |
0 | 2sqrt(0+1)-2 | 0 |
3 | 2sqrt(3+1)-2 | ≈ 1.2 |
5 | 2sqrt(5+1)-2 | ≈ 1.6 |
7 | 2sqrt(7+1)-2 | 2 |
We can now plot the points obtained in the table and connect them with a smooth curve.
Let's now translate this graph 2 units to the right.
Finally, let's reflect the obtained graph in the x-axis.
This graph matches choice A.
We want to obtain the equation of a function whose graph is a translation 2 units to the right followed by a reflection in the x-axis of graph of the given radical function. Let's first write the equation of the translation. To do so, we need to subtract 2 from the function's input. cc Function & Translation2 Units & to the Right y=2sqrt(x+1)-2 & y=2sqrt(x- 2+1)-2 We can simplify the right-hand side of the equation.
Next, let's find the equation that corresponds to a reflection in the x-axis of the translated graph. To do this, we need to multiply the function rule — the function's output — by - 1. cc Function & Reflection in thex -axis y=2sqrt(x-1)-2 & y= - (2sqrt(x-1)-2) We can finally simplify the right-hand side of the last equation by distributing the - 1.