Graph y^3=8x
The problem you are presenting is a mathematical graphing problem. You are asked to visually represent the three-dimensional curve defined by the equation \(y^3 = 8x\) on a coordinate system. This equation describes the relationship between two variables, \(x\) and \(y\). The task typically requires plotting points on the graph that satisfy the equation and drawing a curve that connects these points to show the geometric representation of the algebraic equation.
$y^{3} = 8 x$
Answer
Solution:
Step 1: Determine the value of the function at $x = -1$.
Substitute $x = -1$ into the equation to get $y^3 = 8(-1)$.
Solve for $y$ by taking the cube root of both sides: $y = \sqrt[3]{-8}$.
Simplify to find $y = -2$.
Step 2: Determine the value of the function at $x = 0$.
Substitute $x = 0$ into the equation to get $y^3 = 8(0)$.
Solve for $y$ by taking the cube root of both sides: $y = \sqrt[3]{0}$.
Simplify to find $y = 0$.
Step 3: Determine the value of the function at $x = 1$.
Substitute $x = 1$ into the equation to get $y^3 = 8(1)$.
Solve for $y$ by taking the cube root of both sides: $y = \sqrt[3]{8}$.
Simplify to find $y = 2$.
Step 4: Determine the value of the function at $x = -2$.
Substitute $x = -2$ into the equation to get $y^3 = 8(-2)$.
Solve for $y$ by taking the cube root of both sides: $y = \sqrt[3]{-16}$.
Simplify to find $y = -2\sqrt[3]{2}$.
Step 5: Determine the value of the function at $x = 2$.
Substitute $x = 2$ into the equation to get $y^3 = 8(2)$.
Solve for $y$ by taking the cube root of both sides: $y = \sqrt[3]{16}$.
Simplify to find $y = 2\sqrt[3]{2}$.
Step 6: Graph the function using the calculated points.
Plot the points $(-2, -2.52)$, $(-1, -2)$, $(0, 0)$, $(1, 2)$, and $(2, 2.52)$.
Sketch the curve that passes through these points to represent the function $y^3 = 8x$.
Step 7: Finalize the graph.
Ensure that the curve is smooth and continuous, reflecting the behavior of a cubic function.
Label the axes and the plotted points for clarity.
Knowledge Notes:
The problem involves graphing the cubic function $y^3 = 8x$. The steps taken to graph the function include finding specific points by substituting values of $x$ into the equation and solving for $y$. This is done by taking the cube root of both sides of the equation after multiplying $x$ by 8. The cube root of a number is the value that, when cubed, gives the original number. For example, the cube root of $8$ is $2$ because $2^3 = 8$.
The cube root of a negative number is also negative, as seen in the step where $x = -1$. The cube root of $-8$ is $-2$ because $(-2)^3 = -8$.
When graphing the function, it is important to plot enough points to accurately represent the curve of the cubic function. Cubic functions have an 'S' shaped curve and can have one or two turning points. In this case, the function is not a polynomial but a simple cubic function that can be graphed using the points found.
The final graph should show the behavior of the function, which increases and decreases at different intervals. The plotted points should be connected with a smooth curve that reflects the continuous nature of the function.
