Problem 17 - Calculus Single Variable

Question

Plot several points, and sketch the graph of the function defined by the given expression. $$ \sqrt{2 x+4} $$

Step-by-Step Solution

Answer

Plot points at \((-2, 0), (0, 2), (2, 2.83), (4, 3.46)\); then draw a smooth curve through them.

Step 1: Understanding the Domain

To plot the graph of the function \( f(x) = \sqrt{2x + 4} \), we first need to determine the domain of the function. Since we're dealing with a square root, the expression inside the square root must be greater than or equal to zero. Therefore, we solve the inequality: \( 2x + 4 \geq 0 \). This simplifies to \( x \geq -2 \). Thus, the domain of the function is \( x \geq -2 \).

Step 2: Selecting Key Points

Let's choose several values for \( x \) within the domain and compute corresponding \( f(x) \) values. Common values could be \( x = -2, 0, 2, 4 \). These values will help us understand the shape of the graph.

Step 3: Calculating Function Values

- For \( x = -2 \), \( f(-2) = \sqrt{2(-2) + 4} = \sqrt{0} = 0 \). - For \( x = 0 \), \( f(0) = \sqrt{2(0) + 4} = \sqrt{4} = 2 \). - For \( x = 2 \), \( f(2) = \sqrt{2(2) + 4} = \sqrt{8} \approx 2.83 \). - For \( x = 4 \), \( f(4) = \sqrt{2(4) + 4} = \sqrt{12} \approx 3.46 \).

Step 4: Plotting Points

We can now plot the points \((-2, 0)\), \((0, 2)\), \((2, 2.83)\), and \((4, 3.46)\) on a coordinate plane. These points represent values of the function and help us sketch the overall shape.

Step 5: Sketching the Graph

Draw a smooth curve through the plotted points. The curve will start at the point \((-2, 0)\), curve upwards, revealing the typical shape of a square root function, continuing to rise gently as it extends to the right. Ensure the graph remains on or above the x-axis, as values of \( f(x) \) are non-negative.

Key Concepts

Function Domain

The domain of a function is the set of all possible input values (usually represented by \( x \)) that make the function work. If you think of it like a recipe, the domain includes all the ingredients that you can use without causing the recipe to fail. In the case of a square root function like \( f(x) = \sqrt{2x + 4} \), it's crucial to ensure that whatever we put under the square root produces a non-negative number. This is because square roots of negative numbers aren't defined in the set of real numbers, which is what we typically use in basic graphing.

• To find the domain, solve the inequality \( 2x + 4 \geq 0 \).
• Simplify it to \( x \geq -2 \).

Thus, the function's domain is all real numbers \( x \) greater than or equal to \(-2\). Every input \( x \) must satisfy this condition to keep the function's output valid and real.

Square Root Function

Square root functions, represented generally as \( f(x) = \sqrt{x} \), have a characteristic graph shape. They begin at a certain point on the x-axis (often the origin or some specific starting x-value) and then gently rise to the right as x increases. This rising curve reflects the fact that square roots grow more slowly as numbers increase.

Here are some features of square root functions:

• They begin at the lowest allowable x-value, based on the domain.
• The output or y-value is always non-negative.
• The graph never dips below the x-axis.
• They are not symmetrical and only stretch out in one direction from the start.

For \( f(x) = \sqrt{2x + 4} \), the function starts at \( x = -2 \) where the output is zero, rising smoothly to the right. Understanding its overall shape helps us sketch its graph accurately.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves to visually represent mathematical relationships. It's like a map where every location is identified by a pair of numbers, called coordinates. These coordinates are usually denoted as \((x, y)\), where \( x \) is the horizontal position and \( y \) is the vertical position.

• The horizontal line is called the x-axis.
• The vertical line is called the y-axis.
• The point where the two axes intersect is called the origin, denoted as \((0, 0)\).

When plotting the graph of a function like \( f(x) = \sqrt{2x + 4} \), we use the coordinate plane to display the relationship between x-values and their corresponding y-values. Each pair of calculated \((x, f(x))\) becomes a point on this plane, allowing us to sketch the function's curve and understand how y changes with x.

Plotting Points

Plotting points is the essential step that turns the math behind a function into the visual representation that we can see and analyze on the coordinate plane. Think of it as creating a constellation in a clear sky where each star's position is precisely determined by its coordinates.

• First, compute or calculate key points, which are pairs \((x, f(x))\).
• For example, given \( x \) values such as \(-2, 0, 2, \) and \( 4 \).
• The corresponding \( f(x) \) values are calculated and become y-values.
• Each point \((-2, 0)\), \((0, 2)\), \((2, 2.83)\), and \((4, 3.46)\) sits on the coordinate plane.

Plot these on the plane by placing each point according to its \( x \) and \( y \) coordinates. After plotting, you can connect them with a smooth curve to reveal the overall shape of the function’s graph. This visual process helps in understanding and interpreting how a function behaves over its domain.