Problem 25 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the equation. \(y=\sqrt{x-3}\)

Step-by-Step Solution

Answer

The graph of the equation \(y=\sqrt{x-3}\) starts from the point (3,0) and shoots upwards and to the right. It is undefined for values of x less than 3.

Step 1: Identify the Parent Function

The parent function in this case is \(y=\sqrt{x}\). This function is defined for all \(x\geq 0\) and results in non-negative real numbers.

Step 2: Identify Shifts in the Function

In this function, there is a horizontal shift to the right by \(\mathbf{3}\) units. This is due to the \(x-3\) shift inside the square root function instead of simply \(x\). When \(x=3\), it makes the inside part of the square root equals to \(0\), meaning the curve of the function starts from the point \((3,0)\). For \(x<3\), the function is undefined due to square root of a negative number.

Step 3: Sketch the Graph

Start by plotting the point \((3,0)\) on the graph. This is where the graph starts. Since we know the parent function, it rises slowly and arches upwards to the right in a concave shape. Draw this shape starting from the point \((3,0)\), heading upwards and to the right.

Step 4: Validate with Few More Points

Pick few more values greater than \(3\) to quickly validate the graph. For example when \(x=4\), \(y=\sqrt{4-3}=1\) and when \(x=7\), \(y=\sqrt{7-3}=2\). Add these points \((4,1)\) and \((7,2)\) to the graph to confirm the shape of the function drawn.

Key Concepts

Horizontal Shifts

Understanding horizontal shifts in functions is important for graphing transformations. A horizontal shift occurs when the graph of a function moves left or right on the coordinate plane. With square root functions like \(y=\sqrt{x-3}\), the shift is determined by the expression inside the square root.

In this specific case, \(x-3\) indicates a horizontal shift to the right by 3 units. This is because we replace \(x\) in the parent function \(y=\sqrt{x}\) with \(x-3\). Hence, the graph of the function starts at \((3, 0)\), rather than \((0, 0)\).

Here are some key points to remember about horizontal shifts:

• If the function is \(y=\sqrt{x - h}\), the graph shifts right by \(h\) units.
• If the function is \(y=\sqrt{x + h}\), the graph shifts left by \(h\) units.
• Horizontal shifts do not affect the shape of the graph; they only move it along the x-axis.

Parent Function

The parent function provides the basic shape from which transformations such as shifts, stretches, and reflections occur. For square root functions, the parent function is \(y=\sqrt{x}\).

This simple function forms a curve that starts at \((0, 0)\) and arches upwards and to the right. The domain and range of this function are both restricted to non-negative numbers, as you cannot find the square root of a negative number.

When analyzing transformations, it is helpful to recognize basic characteristics of the parent function:

• The graph begins at the origin \((0, 0)\).
• It continuously increases but never decreases.
• The rate of increase slows down as \(x\) becomes larger.

Fully grasping the parent function allows for easier understanding of how shifts and other transformations alter its graph.

Domain of Function

The domain of a function refers to the set of input values \(x\) for which the function is defined. For square root functions like \(y=\sqrt{x-3}\), the domain is crucial in understanding where the function begins and how it behaves.

In our example, \(y=\sqrt{x-3}\), the expression \(x-3\) inside the square root must be greater than or equal to zero for the function to produce real numbers. Therefore, we solve the inequality \(x-3 \geq 0\) to determine the domain:

• Solving gives \(x \geq 3\), so the domain of \(y=\sqrt{x-3}\) is \([3, \infty)\).
• This indicates the function starts at \(x=3\) and continues indefinitely to the right.

Understanding the domain helps in sketching the graph accurately, knowing the function's starting point, and avoiding undefined regions.