Question

Determine which of the given points are on the graph of the equation. $$ \begin{array}{l} \text { Equation: } y^{2}=x^{2}+9 \\ \text { Points: }(0,3) ;(3,0) ;(-3,0) \end{array} $$

Short answer

The point (0, 3) is on the graph.

Step-by-step solution

Write Down the Equation

The given equation is \[ y^{2} = x^{2} + 9 \]

Check First Point (0, 3)

Substitute \((x, y) = (0, 3)\) into the equation: \[ 3^2 = 0^2 + 9 \] \[ 9 = 9 \] Since the equation holds true, the point \((0, 3)\) is on the graph.

Check Second Point (3, 0)

Substitute \((x, y) = (3, 0)\) into the equation: \[ 0^2 = 3^2 + 9 \] \[ 0 = 9 + 9 \] \[ 0 = 18 \] Since the equation does not hold true, the point \((3, 0)\) is not on the graph.

Check Third Point (-3, 0)

Substitute \((x, y) = (-3, 0)\) into the equation: \[ 0^2 = (-3)^2 + 9 \] \[ 0 = 9 + 9 \] \[ 0 = 18 \] Since the equation does not hold true, the point \((-3, 0)\) is not on the graph.

Key concepts

Coordinate Points

To plot a graph, it's essential to understand coordinate points. A coordinate point is an ordered pair of numbers \((x, y)\), which represent positions in a two-dimensional plane. The first number is the x-coordinate, indicating horizontal position, and the second is the y-coordinate, representing vertical position.
For example, the point (0, 3) means 0 units horizontally and 3 units vertically.
When you are asked to determine if a point lies on the graph of an equation, you substitute these coordinates into the equation to see if the equality holds true.
Let's dive deeper below with more detailed steps using the given equation: \[ y^{2}=x^{2}+9 \].

Substitution Method

The substitution method involves replacing the variables in an equation with given values. This helps to determine if the provided points satisfy the equation.
Here's how you can use it:

• Identify the coordinates \((x,y)\) of the point given.
• Substitute the x-value and y-value into the equation.
• Simplify the resulting expression to see if the left-hand side equals the right-hand side of the equation.

Let's look at the steps:
1. For the point \((0, 3)\), substitute x = 0 and y = 3 into \[ y^{2}=x^{2}+9 \]. Simplifying, you get: \ 3^{2} = 0^{2} + 9 \rightarrow 9 = 9 \.
Since both sides of the equation are equal, the point (0, 3) is on the graph.
2. For the point \((3, 0)\), substitute x = 3 and y = 0. Simplifying, you get: \ 0^{2} = 3^{2} + 9 \rightarrow 0 = 9 + 9 \rightarrow 0 = 18 \.
The left-hand side does not equal the right-hand side, so \((3, 0)\) is not on the graph.
3. Similarly, for \((-3, 0)\), substitute x = -3 and y = 0, resulting in: \ 0^{2} = (-3)^{2} + 9 \rightarrow 0= 9 + 9 \rightarrow 0 = 18 \.
Again, the left-hand side does not equal the right-hand side, proving the point is not on the graph.

Verifying Solutions

Once you substitute the coordinates of a point into the equation, you need to verify if the resulting equality is true. This verifies whether the point lies on the graph.
Verification involves:

• Substituting the x and y values from the point into the equation.
• Simplifying both sides of the equation separately.
• Checking if both sides are equal.

If both sides are equal, the point is a solution to the equation and lies on the graph.

This process can be applied to any point:

• If the simplified equation is true (both sides equal), the point is on the graph.
• If the simplified equation is false (both sides do not equal), the point is not on the graph.

Remember, understanding why a point does or does not satisfy the equation deepens your comprehension of the equation's graph and its solutions.