Problem 6 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether each point lies on the graph of the equation. \(y=\sqrt{5-x}\) (a) (1,2) (b) (5,0)

Step-by-Step Solution

Answer

Both points (1,2) and (5,0) lie on the graph of the equation.

Step 1: Substitution for Point (1,2)

Substitute \(x = 1\) and \(y = 2\) into the equation to see if the point (1,2) satisfies the equation. \[2 = \sqrt{5 - 1}\] Calculate the right side of the equation. If it equals 2 (which is the y-coordinate), then the point lies on the graph

Step 2: Verification for (1,2)

Simplify the right side of the equation: \( \sqrt{4}=2 \). So, \(2=2\). Therefore, the point (1,2) lies on the graph of the equation.

Step 3: Substitution for Point (5,0)

Substitute \(x = 5\) and \(y = 0\) into the equation to see if the point (5,0) satisfies the equation. \[0=\sqrt{5-5}\] Compute the right side of the equation. If it equals 0, which is the y-coordinate, then the point lies on the graph.

Step 4: Verification for (5,0)

Simplify the right side of the equation to \(0=\sqrt{0}=0 \). Therefore, the point (5,0) lies on the graph of the equation.

Key Concepts

Coordinate Substitution

The concept of coordinate substitution is a fundamental technique used in graphing equations. It involves replacing the variables in an equation with the coordinates of a specific point. To see if a point lies on the graph of an equation, we replace the variable 'x' with the x-coordinate of the point and 'y' with the y-coordinate. In our exercise, for instance, we check if the point (1,2) is on the graph of the equation by substituting 'x' with 1 and 'y' with 2.

Here's how it works:

• Take the original equation, in this case, \(y = \sqrt{5 - x}\).
• Replace 'x' with the x-coordinate from the point. Substitute 'y' with the y-coordinate.
• Solve the equation to see if it remains true—meaning both sides of the equation are equal.

If after the substitution the equation holds true, the point is on the graph of the equation. Otherwise, it is not.

Verifying Points on Graphs

Once we have used coordinate substitution, verifying points on a graph is the next step to determine their accuracy on the equation's graph. Verifying is about checking our results to confirm that, after substitution, both sides of the equation are equal.

For example, by substituting the coordinates of point (1,2) into the equation \(y = \sqrt{5 - x}\), we arrived at an equation that simplifies to \(2 = 2\), confirming the point's place on the graph. Similarly, substituting point (5,0) into the equation and confirming that both sides equal zero verifies that this point is also on the graph.

Verification ensures that our substitution is accurate and that the points we graph correspond correctly to the graph of the mathematical function in question.

Radical Equations

Moving beyond the direct application of verifying points on the graph, let's understand radical equations. These are equations that include a variable within a radical—most commonly a square root, as seen in the equation \(y = \sqrt{5 - x}\) from our exercise. Solving radical equations involves isolating the radical on one side of the equation and raising both sides to the power necessary to eliminate the radical.

The concept becomes practical when we deal with points that should or should not lie on the graph of a radical equation. Once the coordinate substitution is done, if the resulting values do not make the equation true (e.g., producing a negative number under the square root), then that point does not lie on the graph of the radical equation.