Problem 46 - Algebra 1: Concepts and Skills

Question

Find the domain of the function. Then sketch its graph and find the range. $$y=\sqrt{x}+4$$

Step-by-Step Solution

Answer

The domain is \(x \geq 0\), the range is \(y \geq 4\), and the graph starts at the point (0,4) and curves upwards to the right.

Step 1 - Finding the Domain

The domain of the function are the values for which the function is defined. To find them, observe that the function is a square root function, hence \(x\) must be greater than or equal to 0. Therefore, the domain is given by \(x \geq 0\).

Step 2 - Sketching the Graph

Start by sketching the graph of \(y = \sqrt{x}\). Because of its nature, it will start at the origin (0,0) and then curve gradually upwards to the right. To sketch \(y = \sqrt{x} + 4\), simply shift the entire original graph upwards by 4 units. The graph will start at the point (0,4) and curve gradually upwards to the right.

Step 3 - Finding the Range

The range of a function are the values that the function can take. By examining the graph, we can observe that \(y\) can be any value greater than or equal to 4. Therefore, the range is \(y \geq 4\).

Key Concepts

Understanding the Square Root Function

The square root function is a fundamental mathematical concept, where for a given non-negative number \(x\), the square root \(\sqrt{x}\) represents a value that, when multiplied by itself, gives \(x\). This function is only defined for \(x \geq 0\) because the square root of a negative number is not real. In the function \(y = \sqrt{x} + 4\), we start with the basic square root function \(y = \sqrt{x}\). Here, 4 is added to the output, which is a transformation of the graph vertically by 4 units upwards. This type of function is often used in various real-world applications, such as in physics to describe wave functions or in statistics for standard deviation calculations.

Graph Sketching Techniques

Graph sketching is a valuable skill in mathematics as it provides a visual representation of a function, helping to understand its properties better. For the function \(y = \sqrt{x}\), the graph typically starts from the origin \((0, 0)\) and rises slowly to the right. It forms a gentle curve due to the nature of the square root's gradual increase.

• Begin by plotting the basic points like \((0, 0)\), \((1, 1)\), \((4, 2)\), and continue to add a few more for precision.
• Connecting these points smoothly will provide the shape of \(y = \sqrt{x}\).

To graph \(y = \sqrt{x} + 4\), notice the transformation involves lifting every point on \(y = \sqrt{x}\) up by 4 units. Thus, the graph begins at \((0, 4)\) instead of at the origin, maintaining its upward trend as before.

Exploring Function Transformations

Function transformation involves changing a function's appearance without altering its core nature. This concept is crucial when modifying existing functions to shift, stretch, or reflect them.

• Vertical transformations involve adding or subtracting a constant to the function's output, which shifts it up or down.
• In \(y = \sqrt{x} + 4\), the '+4' indicates a vertical shift upwards, meaning every output value of \(\sqrt{x}\) has 4 added to it. This does not change the shape, only its position on the graph.

This transformation results in altering the range as well, moving it from \([0, \infty)\) (for \(y = \sqrt{x}\)) to \([4, \infty)\) as seen in \(y = \sqrt{x} + 4\). Understanding these transformations is key in graph manipulations and solving real-world problems involving shifts and scaling.