How to Draw the Graph of x^2 y - sqrt(x) 1

To draw the graph of the equation x^2 y - sqrt(x) 1, we can break it down step-by-step and understand the underlying mathematical principles. Let's start by analyzing the equation in detail.

Step 1: Understand the Equation

The given equation can be viewed as a mathematical relationship involving a term that varies with respect to x. Specifically, the equation can be written as:

x^2 y - sqrt(x) 1

This equation suggests a complex relationship between x and y, and the graph will reflect this relationship.

Step 2: Rearrange the Equation

We can isolate y in terms of x to better understand the equation:

y 1 - x^2 sqrt(x)

However, we can further decompose it into two separate equations based on the nature of the square root:

y - sqrt(x) ± sqrt(1 - x^2)

This gives us two possible functions:

Upper Function: y sqrt(x) sqrt(1 - x^2) Lower Function: y sqrt(x) - sqrt(1 - x^2)

Step 3: Determine the Domain

The term sqrt(1 - x^2) indicates that the value inside the square root must be non-negative, hence:

1 - x^2 ≥ 0

This leads to the domain:

-1 ≤ x ≤ 1

Step 4: Analyze the Functions

Now that we have the functions, we can analyze them within the given domain:

For x ≥ 0

Upper Function: y sqrt(x) sqrt(1 - x^2) Lower Function: y sqrt(x) - sqrt(1 - x^2)

For x

Upper Function: y sqrt(-x) sqrt(1 - x^2) Lower Function: y sqrt(-x) - sqrt(1 - x^2)

These two functions will describe the upper and lower curves of the graph respectively.

Step 5: Graphing the Functions

Let's plot some specific points to help visualize the graph:

x 0: Upper Function: y sqrt(0) sqrt(1 - 0^2) 1 Lower Function: y sqrt(0) - sqrt(1 - 0^2) -1 x 1: Upper Function: y sqrt(1) sqrt(1 - 1^2) 1 Lower Function: y sqrt(1) - sqrt(1 - 1^2) 1 x -1: Upper Function: y sqrt(-1) sqrt(1 - (-1)^2) (Note: This part is complex as sqrt(-1) is not real) Lower Function: y sqrt(-1) - sqrt(1 - (-1)^2) (Similarly, this part is not real) x 0.5: Upper Function: y sqrt(0.5) sqrt(1 - 0.5^2) sqrt(0.5) sqrt(0.75) Lower Function: y sqrt(0.5) - sqrt(0.75)

Step 6: Sketch the Graph

Using these points, we can plot the graph and sketch the curves. The graph will show symmetry about the y-axis, with two curves above and below each other, reflecting the upper and lower functions described earlier.

Summary

The graph of the equation x^2 y -sqrt(x) 1 will depict two curves for x in the range [-1, 1] with the upper curve above y 1 and the lower curve below y -1.

Related Keywords: graph equation, mathematical graph, algebraic graph