Graph The Function \[$ F(x) = \sqrt[3]{x} - 5 \$\].Plot Five Points On The Graph Of The Function As Follows:- Plot The First Point Using The \[$ X \$\]-value That Satisfies \[$ \sqrt[3]{x} = 0 \$\].- Plot Two Points To The Left
Understanding the Function
The given function is { f(x) = \sqrt[3]{x} - 5 $}$. This is a cubic root function, which means that the output of the function is the cube root of the input value. The function is then subtracted by 5, resulting in a shifted cubic root function.
Graphing the Function
To graph the function, we need to understand its behavior. The cubic root function { \sqrt[3]{x} $}$ is an increasing function, meaning that as the input value increases, the output value also increases. However, the function is shifted down by 5 units, resulting in a graph that is 5 units below the x-axis.
Plotting Points
We are asked to plot five points on the graph of the function. The first point is to be plotted using the { x $}$-value that satisfies { \sqrt[3]{x} = 0 $}$. This means that we need to find the value of x that makes the cubic root of x equal to 0.
Finding the Value of x
To find the value of x, we can set up the equation { \sqrt[3]{x} = 0 $}$ and solve for x. Since the cube root of x is equal to 0, we can raise both sides of the equation to the power of 3 to get rid of the cube root.
{ \sqrt[3]{x} = 0 $}{$ x = 0^3 $}{$ x = 0 $}$
Therefore, the value of x that satisfies the equation is 0.
Plotting the First Point
Now that we have found the value of x, we can plot the first point on the graph. The first point is plotted at the point (0, -5), since the function is shifted down by 5 units.
Plotting Two Points to the Left
We are asked to plot two points to the left of the first point. To do this, we need to find two values of x that are less than 0 and plot the corresponding points on the graph.
Finding the Values of x
To find the values of x, we can use the equation { \sqrt[3]{x} = -5 $}$ and solve for x. Since the cube root of x is equal to -5, we can raise both sides of the equation to the power of 3 to get rid of the cube root.
{ \sqrt[3]{x} = -5 $}{$ x = (-5)^3 $}{$ x = -125 $}$
Therefore, the value of x that satisfies the equation is -125.
Plotting the Second Point
Now that we have found the value of x, we can plot the second point on the graph. The second point is plotted at the point (-125, 0), since the function is shifted down by 5 units.
Finding the Third Value of x
To find the third value of x, we can use the equation { \sqrt[3]{x} = -10 $}$ and solve for x. Since the cube root of x is equal to -10, we can raise both sides of the equation to the power of 3 to get rid of the cube root.
{ \sqrt[3]{x} = -10 $}{$ x = (-10)^3 $}{$ x = -1000 $}$
Therefore, the value of x that satisfies the equation is -1000.
Plotting the Third Point
Now that we have found the value of x, we can plot the third point on the graph. The third point is plotted at the point (-1000, 5), since the function is shifted down by 5 units.
Finding the Fourth Value of x
To find the fourth value of x, we can use the equation { \sqrt[3]{x} = -15 $}$ and solve for x. Since the cube root of x is equal to -15, we can raise both sides of the equation to the power of 3 to get rid of the cube root.
{ \sqrt[3]{x} = -15 $}{$ x = (-15)^3 $}{$ x = -3375 $}$
Therefore, the value of x that satisfies the equation is -3375.
Plotting the Fourth Point
Now that we have found the value of x, we can plot the fourth point on the graph. The fourth point is plotted at the point (-3375, 10), since the function is shifted down by 5 units.
Finding the Fifth Value of x
To find the fifth value of x, we can use the equation { \sqrt[3]{x} = -20 $}$ and solve for x. Since the cube root of x is equal to -20, we can raise both sides of the equation to the power of 3 to get rid of the cube root.
{ \sqrt[3]{x} = -20 $}{$ x = (-20)^3 $}{$ x = -8000 $}$
Therefore, the value of x that satisfies the equation is -8000.
Plotting the Fifth Point
Now that we have found the value of x, we can plot the fifth point on the graph. The fifth point is plotted at the point (-8000, 15), since the function is shifted down by 5 units.
Conclusion
Q: What is the cubic root function?
A: The cubic root function is a mathematical function that takes a number as input and returns the cube root of that number. It is denoted by the symbol { \sqrt[3]{x} $}$.
Q: How do you graph the cubic root function?
A: To graph the cubic root function, you can start by plotting the point (0, 0) on the coordinate plane. Then, you can plot additional points by raising the x-coordinate to the power of 3 and taking the cube root of the result.
Q: What is the difference between the cubic root function and the square root function?
A: The main difference between the cubic root function and the square root function is the power to which the input value is raised. The square root function raises the input value to the power of 1/2, while the cubic root function raises the input value to the power of 1/3.
Q: How do you shift the cubic root function down by 5 units?
A: To shift the cubic root function down by 5 units, you can subtract 5 from the output of the function. This can be represented mathematically as { f(x) = \sqrt[3]{x} - 5 $}$.
Q: What is the value of x that satisfies the equation { \sqrt[3]{x} = 0 $}$?
A: The value of x that satisfies the equation { \sqrt[3]{x} = 0 $}$ is 0.
Q: How do you plot the first point on the graph of the function?
A: To plot the first point on the graph of the function, you can use the value of x that satisfies the equation { \sqrt[3]{x} = 0 $}$. In this case, the first point is plotted at the point (0, -5).
Q: How do you plot two points to the left of the first point?
A: To plot two points to the left of the first point, you can use the equation { \sqrt[3]{x} = -5 $}$ and solve for x. This will give you two values of x that are less than 0, which you can then use to plot the corresponding points on the graph.
Q: What is the value of x that satisfies the equation { \sqrt[3]{x} = -5 $}$?
A: The value of x that satisfies the equation { \sqrt[3]{x} = -5 $}$ is -125.
Q: How do you plot the remaining points on the graph?
A: To plot the remaining points on the graph, you can use the same process as before. You can use the equation { \sqrt[3]{x} = -10 $}$ and solve for x to get the value of x that satisfies the equation. Then, you can plot the corresponding point on the graph.
Q: What is the value of x that satisfies the equation { \sqrt[3]{x} = -10 $}$?
A: The value of x that satisfies the equation { \sqrt[3]{x} = -10 $}$ is -1000.
Q: How do you plot the final point on the graph?
A: To plot the final point on the graph, you can use the same process as before. You can use the equation { \sqrt[3]{x} = -20 $}$ and solve for x to get the value of x that satisfies the equation. Then, you can plot the corresponding point on the graph.
Q: What is the value of x that satisfies the equation { \sqrt[3]{x} = -20 $}$?
A: The value of x that satisfies the equation { \sqrt[3]{x} = -20 $}$ is -8000.
Conclusion
In conclusion, we have graphed the function { f(x) = \sqrt[3]{x} - 5 $}$ and plotted five points on the graph. We have also answered some common questions about the cubic root function and how to graph it.