Graph the following function using transformations. Be sure to graph all of the stages on one graph. State the domain and range. y = -SQRT x - 6.
My instructor is saying that I have to do more than one graph. Can anybody help me I am lost!
I will assume you mean
y = -√(x-6) and NOT y = -(√x) - 6
start with y = √x
using the following points, sketch the graph.
(0,0), (1,1), (4,2), (9,3), (16,4) ...
(notice I pick numbers that work out nicely)
You will have a "swooping" curve in the first quadrant that rises slowly.
now sketch y = - √x by reflecting your graph in the x-axis, (just flip it downwards)
then move the second graph 6 units to the RIGHT
(move each point 6 units to the right)
domain: any value of x , x ≥ 6
range: any real value of y , y ≤ 0 (all below the x-axis)
To verify, enter
(x-6)^.5
in the "first graph" in
http://rechneronline.de/function-graphs/
change "Range x-axis from" 6 to 20
Sure, I can help you with graphing the function. To graph the function y = -√(x - 6), we can start by graphing the basic function y = √x.
The basic function y = √x starts at the point (0, 0) and increases as x increases. The domain of this function is x ≥ 0, and the range is y ≥ 0.
To graph the function y = -√(x - 6), we can apply the following transformations to the basic function:
1. Horizontal shift: The "- 6" inside the square root function indicates that the graph is shifted 6 units to the right. So, the new graph will start at the point (6, 0), and increase as x increases. The domain remains the same: x ≥ 0.
2. Reflect: The negative sign in front of the square root function reflects the graph across the x-axis. So, the graph will now decrease as x increases. The range also changes to y ≤ 0.
Combining these transformations, we can graph the function y = -√(x - 6) as follows:
1. Start by graphing the basic function y = √x.
2. Shift the graph 6 units to the right.
3. Reflect the graph across the x-axis.
The final graph will start at the point (6, 0), and decrease as x increases. The domain remains the same: x ≥ 0, and the range is y ≤ 0.
Sure, I can help you with that!
To graph the function y = -√(x-6), we can start by understanding the transformations involved.
1. Horizontal Shift: The function inside the square root, x-6, indicates a horizontal shift of 6 units to the right. So, the graph will be shifted right by 6 units compared to the standard square root function.
2. Vertical Shift: The negative sign in front of the square root function signifies a vertical reflection, which means the graph will be reflected downwards about the x-axis.
3. Stretch/Compression: There is no number directly multiplying the square root function, indicating a vertical stretch or compression. So, the graph will have the same slope or vertical stretch as the standard square root function.
Now let's break it down step by step:
Step 1: Graph the standard square root function y = √x:
To plot the standard square root function, you can start by selecting some x-values (such as 0, 1, 4, 9, etc.) and finding their corresponding y-values by taking the square root of each x-value.
For example:
When x = 0, y = √0 = 0.
When x = 1, y = √1 = 1.
When x = 4, y = √4 = 2.
When x = 9, y = √9 = 3.
Plotting these points, you will get the graph of y = √x as a curve starting from the origin and extending towards the positive y-axis.
Step 2: Apply the Transformations:
a. Horizontal Shift: To shift the graph 6 units to the right, add 6 to each x-value calculated in step 1.
For example:
When x = 0 + 6 = 6, y = √0 = 0.
When x = 1 + 6 = 7, y = √1 = 1.
When x = 4 + 6 = 10, y = √4 = 2.
When x = 9 + 6 = 15, y = √9 = 3.
Plot these points on your graph, and you will notice that the graph has shifted to the right by 6 units.
b. Vertical Reflection: Apply the vertical reflection by negating the y-values calculated in step 2a.
For example:
When x = 6, y = - √6 = -0.
When x = 7, y = - √1 = -1.
When x = 10, y = - √2 = -2.
When x = 15, y = - √3 = -3.
Plotting these reflected points on your graph, you will see that the graph is now reflected downwards.
Step 3: Determine the Domain and Range:
The domain of the function y = -√(x-6) is determined by the values of x for which the function is defined. Since we're taking the square root of the expression (x-6), we need to ensure that (x-6) is greater than or equal to zero to avoid imaginary numbers. Therefore, the domain is x ≥ 6.
The range of the function is the set of all possible y-values the function takes. Since the graph is reflected downward, the range will be all y-values less than or equal to zero. So, the range is y ≤ 0.
Now that you have completed all the stages, plot the transformed points on the same graph as the standard square root function, connecting them to form a curved line. This will give you the final graph of y = -√(x-6).
I hope that helps! Let me know if you have any further questions.