Problem 71 - Algebra 2

Question

How is the graph of \(y=\sqrt{x+7}\) translated from the graph of \(y=\sqrt{x} ?\) A. shifted 7 units left B. shifted 7 units right C. shifted 7 units up D. shifted 7 units down

Step-by-Step Solution

Answer

A. shifted 7 units left

Step 1: Understand Graph Translations

An essential principle to know is transformations in function graphs. When an adjustment is done inside the function, with x itself, it results in horizontal shifts. When a value is added to x, it shifts the graph towards the left side, and when it is subtracted, it shifts the graph towards the right side.

Step 2: Analysing the Given Function

Looking at the function \(y=\sqrt{x+7}\), what we have here is an 'addition of 7' with x INSIDE the square root function. This indicates a horizontal translation of the graph.

Step 3: Determine the Direction of Shift

Since there is 'addition of 7' inside the function with x, the graph of the function \(y=\sqrt{x}\) will shift horizontally towards the left by 7 units.

Key Concepts

Square Root Functions

Square root functions are a fundamental part of mathematics and can be commonly seen in graphs as half-parabolas that start at a certain point and extend infinitely in one direction. The parent function, usually represented as \( y = \sqrt{x} \), serves as the basic version of these functions. This graph starts at the origin point \((0,0)\) and extends to the right, along the x-axis. It's worth noting that, for the parent function, the domain is \([0, \infty)\) and the range is also \([0, \infty)\). This tells us which x-values can be used in the function, and what y-values will come out.

As the function develops through transformations, its shape remains a characteristic curve but its position or orientation can change. Understanding square root functions helps in comprehending how these transformations impact a graph's position.

Function Transformations

Function transformations encompass a set of operations that systematically alter the appearance of the graph of a function. For square root functions like \( y = \sqrt{x} \), these transformations usually include translations, reflections, stretch, or compressions. Translations involve shifting the entire graph either vertically or horizontally, while reflections may flip the graph over the x or y-axis.

When examining the transformation in the function \( y = \sqrt{x+7} \), we notice a horizontal shift occurring, which is a form of translation. Depending on whether a number is added or subtracted from 'x' within the function, this determines the direction of the shift. Additive transformations in a function alter the graph’s position without changing its shape or orientation.

Horizontal Shifts

Horizontal shifts are one of the most common transformations applied to square root functions. They occur when a number is added or subtracted from the variable 'x' inside the function itself. When a positive number is added, like in the function \( y = \sqrt{x+7} \), the graph shifts to the left by that number of units, in this case, 7 units. Conversely, subtracting a positive number would direct the shift to the right.

Understanding the logic behind horizontal shifts helps in predicting graph movements accurately. For the function \( y = \sqrt{x+7} \), knowing that we add '7' to 'x' inside the square root function sets the expectation for a leftward shift. This knowledge is essential not only for solving problems but also for drawing accurate graphs and analyzing real-world data.