Problem 5 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether each point lies on the graph of the equation. \(y=\sqrt{x+4}\) (a) (0,2) (b) (12,4)

Step-by-Step Solution

Answer

Yes, both points (0,2) and (12,4) lie on the graph of the equation \(y=\sqrt{x+4}\).

Step 1: Investigate point (0,2)

Substitute \(x=0\) and \(y=2\) into \(y=\sqrt{x+4}\). If the equation is satisfied, then the point lies on the graph, otherwise it does not. \nSo we will substitute and see if the values match as follows: \( 2=\sqrt{0+4}\). Which after simplifying gets \(2 = \sqrt{4}\). Which means that \(2=2\). The equation holds true. Hence, the point (0,2) lies on the graph of the equation.

Step 2: Investigate point (12,4)

Substitute \(x=12\) and \(y=4\) into \(y=\sqrt{x+4}\). If the equation is satisfied, then the point lies on the graph, otherwise it does not. \nSo we substitute and check if the values match as follows: \(4=\sqrt{12+4}\). Which simplifies to \(4 = \sqrt{16}\). Meaning that \(4 = 4\). This equation also holds true. Hence, the point (12,4) also lies on the graph of the equation.

Key Concepts

Square Roots

Square roots are a fundamental mathematical concept that involve finding a number which, when multiplied by itself, returns the original value. For example, the square root of 9 is 3, because 3 times 3 is 9.
When dealing with graphs, square roots are often involved in equations that describe the curve or line you want to study. In our example equation, \[y = \sqrt{x+4}\]we are looking at how the value of \(y\) changes as \(x\) changes.
The key thing to remember with square roots is they can never result in negative values when dealing with real numbers. This is because a negative number cannot be squared to get a positive number, thus you won't see negative values for \(y\) in graphs involving basic square root functions. In practical terms, when checking if a point lies on a square root graph, substitute \(x\) into the equation, and see if the resulting \(y\) matches the \(y\) of the point in question.

Coordinate System

The coordinate system is like a grid that helps us locate points in a plane. This system is based on two axes: the horizontal axis, called the x-axis, and the vertical axis, known as the y-axis. Each axis divides the plane into four quadrants.
Each point in this system is identified by an \((x, y)\) pair. The first number corresponds to the x-coordinate, and the second to the y-coordinate.

• The point \((0, 0)\), called the origin, is where both axes cross.
• A positive x-value indicates a point is to the right of the origin, while a negative x-value places it to the left.
• Positive y-values place a point above the origin; negative y-values place it below.

When determining if a point lies on a graph described by an equation, substitute the point's x-value into the equation and check if the resulting y-value matches the point's y-coordinate.

Substitution Method

The substitution method involves replacing variables with specific values to solve equations or check solutions.
To determine if a point lies on the graph of an equation, substitute the \(x\) and \(y\) values from the point into the equation.

• If the equation holds true, then the point lies on the graph.
• If the equation does not hold true, then the point is not on the graph.

For example, given the point \((12, 4)\) and the equation \[y = \sqrt{x+4}\]you would substitute \(x = 12\) and \(y = 4\):\[4 = \sqrt{12 + 4}\]This simplifies to \[4 = \sqrt{16}\], which is true because the square root of 16 is indeed 4. Thus, the point lies on the graph.
Substituting values is a simple yet powerful method to verify solutions and comprehend how equations behave with different input values.