Simplified: Domain, Range And Graph Of Sqrt(3X + 18) | Algebra, Math Graphs

y = (3x + 18)^(1/2)

y’ = 1.5(3x + 18)^(-1/2)

To work with the equation y = (3x + 18)^(1/2), we can use a few algebraic steps to simplify it, find its domain and range, and sketch its graph.

1. Simplifying the equation:
We can begin by rewriting the equation as follows:

y^2 = (3x + 18)

Now we have an equation that can be more easily manipulated. Let’s solve for x:

y^2 – 18 = 3x

x = (y^2 – 18) / 3

This is a simplified form of the equation. We can see that it is a function of y, which means that for each value of y, there is only one corresponding value of x.

2. Finding the domain and range:
The domain of a function is the set of all possible input values, while the range is the set of all possible output values. In this case, we need to determine what values of x and y make sense within the context of the problem.

For the square root function, the only restriction is that the argument (the expression inside the parentheses) must be nonnegative. Therefore, we must have:

3x + 18 >= 0

Solving for x:

x >= -6

So the domain of the function is all real numbers greater than or equal to -6.

For the range, we can see that the square root of any nonnegative number is always nonnegative. So the range of the function is all nonnegative real numbers.

3. Sketching the graph:
To sketch the graph, we can use the simplified form:

x = (y^2 – 18) / 3

This is a quadratic equation in y, which means that the graph is a parabola. The coefficient of y^2 is 1/3, which means that the parabola opens upward and is relatively narrow.

To plot the graph, we can choose a few values of y, plug them into the equation, and find the corresponding values of x. Then we can connect the points with a smooth curve.

For example, when y = 0:

x = (0^2 – 18) / 3 = -6

So one point on the graph is (-6, 0).

When y = 1:

x = ((1)^2 – 18) / 3 = -17/3

Another point on the graph is (-17/3, 1).

Similarly, we can find more points and sketch the graph. Here is a rough sketch of the graph:

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The vertex of the parabola is at (-6, 0), and the axis of symmetry is the vertical line x = -6. The graph increases without bound as y approaches infinity.

Overall, the equation y = (3x + 18)^(1/2) represents a square root function with a domain of all real numbers greater than or equal to -6 and a range of all nonnegative real numbers. The graph is a parabola that opens upward and is centered at (-6, 0).

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