Problem 15 - Algebra 2

Question

Graph each function. \(y=-\sqrt{x-3}+2\)

Step-by-Step Solution

Answer

The graph of the function \(y = -\sqrt{x-3} + 2\) starts at the point (3,2), extends only to the right as \(x\) is greater than or equal to 3, and moves downwards because of the negative sign in front of the square root. It looks like a mirrored, translated square root graph.

Step 1: Understanding the square root function

A basic square root function, \(y = \sqrt{x}\), starts at the origin (0,0) and then increases as x increases. This graph is always positive and always increasing.

Step 2: Translating the graph

In our function, we have \(y = -\sqrt{x-3} + 2\), which is a square root function that has been translated to the right by 3 units (due to \(x-3\)) and translated upwards by 2 units (due to \(+2\)). Note that the minus sign in front of the square root sign also flips the square root graph downwards.

Step 3: Identifying the range of x-values

Since the square root of a negative number is undefined, for \(x - 3\) to be greater than or equal to zero, \(x\) must be greater than or equal to 3. Therefore, the x-values will have a range of [3,∞).

Step 4: Sketching the graph

We can now sketch the graph, starting at the point (3,2) and extending only to the right (positive x-values) and downwards (because of the negative sign). The graph will look like a mirrored, translated square root graph.

Step 5: Checking points

To ensure the graph has been sketched correctly, check a few points that satisfy the function. These could, for example, be (4, -1), (5, -√2) and (6, -2).

Key Concepts

Translation of Functions

Translation of functions is a mathematical process that shifts a graph horizontally or vertically on the coordinate plane without changing its shape or orientation. Consider the function \(y = -\sqrt{x-3} + 2\). Here, the expression \(x-3\) indicates a horizontal translation.

• The value of \(-3\) suggests that the graph moves to the right by 3 units.
• The \(+2\) outside the square root moves the entire graph up by 2 units.

This type of transformation allows us to adjust where the graph sits without altering the graph's original form or size; it only shifts its position.

Reflection of Functions

A reflection changes the direction of a graph across an axis, which can result in a mirror image of the original graph. In our function \(y = -\sqrt{x-3} + 2\), the negative sign before the square root is crucial.

• This negative sign reflects the graph over the x-axis.
• While a standard square root function like \(y = \sqrt{x}\) increases, our reflected function decreases.

This means that unlike the standard square root curve, which curves upwards, the graph of this function will curve downwards, starting from the point of translation.

Domain and Range in Functions

The domain and range of a function define the set of input and output values, respectively. For the function \(y = -\sqrt{x-3} + 2\), calculating the domain involves setting constraints on \(x\).

• The term \(x-3\) under the square root dictates that \(x\) must be greater than or equal to 3 to keep the expression real and defined. Thus, the domain is \([3, \infty)\).
• The range is determined by how the function values behave. Given the reflection, the highest point is at \(y=2\), decreasing indefinitely as \(x\) increases, establishing a range of \((-\infty, 2]\).

Sketching Graphs

Sketching graphs of functions involves plotting key points and interpreting the overall shape and behavior of the graph. Following the previous transformations, you can sketch \(y = -\sqrt{x-3} + 2\) with the following steps:

• Identify the starting point at \((3,2)\), by applying the translations to the basic parent function.
• Due to the reflection, sketch the graph moving downwards towards the right.
• Plot additional points by calculating the function's value for various \(x\), such as \((4, -1)\), \((5, -\sqrt{2})\), and \((6, -2)\).

The sketch should depict the downward-sloping, right-moving nature of the translated and reflected square root function.