Problem 54 - Algebra 2

Question

a. The graph of \(y=\sqrt{x}\) is translated five units to the right and two units down. Write an equation of the translated function. b. The translated graph from part (a) is again translated, this time four units left and three units down. Write an equation of the translated function.

Step-by-Step Solution

Answer

a. The equation of the translated function is \(y=\sqrt{x+5}-2\).\nb. The equation of the translated function is \(y=\sqrt{x+1}-5\).

Step 1: First Translation

To translate the graph of \(y=\sqrt{x}\) five units to the right and two units down, replace \(x\) with \(x+5\) and \(y\) with \(y+2\) in the equation \(y=\sqrt{x}\). Hence, the equation of the translated function is \(y=\sqrt{x+5}-2\).

Step 2: Second Translation

Now we have to translate the graph we obtained in step 1 that is \(y=\sqrt{x+5}-2\), four units to the left and three units down. So, replace \(x\) with \(x-4\) and \(y\) with \(y+3\) in the equation \(y=\sqrt{x+5}-2\). Therefore, the equation of the translated function is \(y=\sqrt{x+1}-5\).

Key Concepts

Translation of Functions

A function translation is a method to shift the graph of a function along the coordinate plane. When you translate a function, you're essentially moving its graph to a new location without altering its shape or orientation. To perform such translations:

• For a horizontal translation, replace every instance of the variable \(x\) in the function with \(x-h\), where \(h\) is the number of units to move right (if positive) or left (if negative).
• For a vertical translation, replace every instance of \(y\) with \(y-k\), where \(k\) is the number of units to move up (if positive) or down (if negative).

Let's consider how this applies to our original function \(y = \sqrt{x}\). - To translate it five units to the right and two units down, we applied these changes to get \(y = \sqrt{x+5} - 2\). - Repeating the process moving four units left and three down results in \(y = \sqrt{x+1} - 5\). Thus, you can see how these transformations translate the graph's position on the plane.

Radical Functions

Radical functions involve roots, such as square roots, cube roots, etc. They play an essential role in mathematics, especially when modeling phenomena involving rates of change.For example, the function \(y = \sqrt{x}\) is a basic radical function. This function starts at \(x = 0\) and increases as \(x\) increases. It is defined for all non-negative \(x\) values, and its graph is a curve that rises slowly and steadily.Radical functions have a few key characteristics:

• They are usually not defined for negative values under the root for even roots, such as square roots.
• The domain of a basic radical function like \(y = \sqrt{x}\) is \(x \geq 0\).
• Visually, they form a curved shape known as a radical curve.

Understanding these facets helps in transforming and manipulating equations involving radical functions.

Graphing Functions

Graphing functions is a crucial skill in math that allows you to visualize relationships between variables. It involves plotting points on a coordinate grid based on a given equation and then connecting those points to form the graph of the function.Here's a step-by-step on how to graph \(y = \sqrt{x}\):1. **Identify the Domain:** Since \(y = \sqrt{x}\) is not defined for negative \(x\), the domain is \(x \geq 0\).2. **Choose Points:** Select a few points within the domain. For example, \((0,0)\), \((1,1)\), and \((4,2)\) because \(\sqrt{1} = 1\), \(\sqrt{4} = 2\).3. **Plot Points:** Plot these chosen points on the graph.4. **Draw the Curve:** Connect the points smoothly to display the curve.Transforming the base function \(y = \sqrt{x}\) to \(y = \sqrt{x+5} - 2\) involves:

• **Shift Right:** Move the entire curve 5 units to the right.
• **Shift Down:** Then, move the curve 2 units downward.

By plotting the points from your translated function equation, you can see how the graph evolves across transformations. Visualizing the effects makes understanding translations and transformations insightful and intuitive.