Problem 17 - Algebra 1: Concepts and Skills

Question

Find the domain of the function. Then sketch its graph. $$y=3 \sqrt{x+1}$$

Step-by-Step Solution

Answer

The domain of the function \(y = 3\sqrt{x+1}\) is \(x \geq -1\). After plotting key points and sketching the graph, it is observed that the graph is an upward opening curve starting from point (-1, 0). The curve follows along the key points and showcases typical characteristics of the square root function.

Step 1: Find the Domain of the Function

The value within the square root, \(x+1\), cannot be negative. Hence, we need to solve for \(x\) in the inequality: \[x + 1 \geq 0\] Subtracting 1 from both sides gives: \[x \geq -1\] This is the domain of the function.

Step 2: Find Key Points for Graphing

To sketch the graph, we should find the square root function's key points. Choosing representative values for \(x\) in the domain, that is, \(x = -1, 0, 1, 2, 3,\). Then proceeding to find the corresponding \(y\) values by substituting these \(x\) values back into the function \(y = 3\sqrt{x+1}\), we get the following points: (-1, 0), (0, 3), (1, 6), (2, 6.93), and (3, 7.74).

Step 3: Graph the Function

The graph of the function can be drawn using the points derived in the above step. Start at the point (-1, 0), this is the point where sqrt{0} = 0, then graph passes through the other points and continues along the curve to the right. The curve starts at the point (-1, 0), and as the x-values increase, the y-values also increase. The graph is an upward opening curve starting from point (-1, 0), demonstrating typical characteristics of the square root function. Note that it does not extend into the left of (-1, 0) because the function is not defined there, as per the domain.

Key Concepts

Square Root Function

The square root function is a significant function in mathematics often encountered in various problems. It is defined as the function \( y = \sqrt{x} \), where \( x \geq 0 \). The function returns the non-negative square root of \( x \).
For our exercise, the square root function is modified to include the expression \( x + 1 \) under the square root, leading to \( y = 3 \sqrt{x+1} \). This modification shifts the starting point of the function to the left on the x-axis.

• It is essential to remember that the expression inside the square root must be non-negative.
• This condition ensures that the function always yields a real number output.
• This is found by solving \( x+1 \geq 0 \), or \( x \geq -1 \), to find the domain of the function.

The result creates a "starting point" at \( x = -1 \), hence any \( x \) input less than \(-1\) is invalid for this function.

Graphing Calculators

Graphing calculators are excellent tools for visualizing mathematical functions. These devices or computer applications allow students to input mathematical expressions and view their corresponding graphs. Using a graphing calculator to sketch \( y = 3\sqrt{x+1} \)

• helps students see the relationship between the x-values and their associated y-values,
• and understand how inputs affect outputs graphically.
• This visual approach makes it easier to grasp concepts like domain and range.

When dealing with square root functions, graphing calculators can highlight how changes and shifts in parameters, such as the transformation from \( \sqrt{x} \) to \( \sqrt{x+1} \), impact the graph's shape and position on the coordinate plane.

Inequalities

Inequalities play a key role in finding the domain of functions, especially those involving square roots. Inequalities are expressions that define a range of acceptable values for variables.
In the context of our square root function, the inequality \( x + 1 \geq 0 \) ensures that the input inside the square root is non-negative.

• This inequality tells us that \( x \) must be greater than or equal to \(-1\) for the function to have real number outputs.
• By solving this inequality, we determine the domain, which for this function is all real numbers \( x \geq -1 \).

Understanding inequalities allows students to solve real-life math problems where certain limits or boundaries are essential to consider. Inequalities are not only crucial in mathematics but also apply to numerous scientific fields, offering a foundational skill set in problem-solving.

Function Graphing

Graphing functions helps to see how mathematical equations behave visually. In our specific example of \( y = 3 \sqrt{x+1} \), graphing involves plotting points that satisfy this equation and connecting them smoothly.

• Start by identifying key points using the found domain, \(-1\), as the starting point.
• Calculate corresponding y-values for chosen x-values that satisfy \( x \geq -1 \).
• The points \((-1, 0), (0, 3), (1, 6), (2, 6.93), (3, 7.74)\) are plotted.
• The curve passes through these points, indicating how the function grows as \( x \) increases.

In general, graphing helps students understand not just where a function starts and stops, but also the rate and direction at which it changes. Graphical representation provides clear insights into function behavior that algebraic manipulation alone might not reveal.