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The distance formula is a crucial concept in geometry that helps in determining the distance between two points in the Cartesian plane. It is derived from the Pythagorean theorem and is given by the formula:\[L = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.\] This formula allows us to calculate how far apart two locations, represented by their coordinates \((x_1, y_1)\) and \((x_2, y_2)\), are from each other.

• This formula is handy for all sorts of calculations in math and physics.
• Understanding it can help you in fields like engineering and computer graphics.

Applying the distance formula begins with identifying the coordinates of the points you need to evaluate, which in our exercise were \((x, y)\) and \((4, 0)\). By substituting these into the formula, you find the expression for the distance, \[L = \sqrt{(4 - x)^2 + y^2}.\] Simple enough, right? Remember, practice helps make this process more intuitive.

In many mathematical problems, expressing one quantity as a function of another can simplify analysis. In this case, the problem requires that we express the distance \(L\) as a function of \(y\). A function of a variable like \(y\) expresses how \(L\) changes in response to changes in \(y\).

• Start by identifying the relationship between the variables involved.
• Use the given equations or relationships to isolate your variable of interest.

Here, given the point \((x, y)\) is on the graph of \(y = \sqrt{x - 3}\), we can swap roles of \(x\) and \(y\) using algebra to find \(x\) in terms of \(y\). Doing this, we derive:\[x = y^2 + 3.\] This expression lets us substitute \(x\) in the formula for \(L\), turning the distance into a function purely of the variable \(y\). That makes analyzing how changes in \(y\) affect the distance much more straightforward.

Visualizing functions on a graph can provide great insights into their behavior. In our case, the function \(y = \sqrt{x - 3}\) is a graph that helps us relate \(y\) and \(x\). This graph represents a curve that starts from the point \((3, 0)\) and continues to the right, moving upwards.

• The graph represents all the possible \((x, y)\) pairs that satisfy the equation.
• Understanding the shape of this graph is key to understanding the problem itself.

By knowing that \(y\) must be positive (because it's under a square root), the graph of \(y = \sqrt{x - 3}\) only includes points above the x-axis. This restriction affects how we manipulate the equation and visualize the changes in \(L\) as a function of \(y\). Visual aids like graphs can greatly enhance your comprehension!

Algebraic manipulation involves rearranging equations and expressions to solve or simplify them. It is a foundational skill in mathematics. In our exercise, we needed to solve for \(x\) in terms of \(y\) from the original function \(y = \sqrt{x - 3}\).

• First, square both sides to eliminate the square root: \(y^2 = x - 3\).
• Next, rearrange the equation to isolate \(x\): \(x = y^2 + 3\).

After obtaining \(x = y^2 + 3\), we substituted this into the distance formula \(L\). This manipulation simplified the originally complex expression, making it more manageable:\[L = \sqrt{(1 - y^2)^2 + y^2} = \sqrt{1 - y^2 + y^4}.\]It demonstrates the power of algebraic manipulation to convert problems into forms more amenable to analysis.