Problem 7 - Algebra 2

Question

Graph each function. \(y=\sqrt{x+6}\)

Step-by-Step Solution

Answer

The graph of the function \(y = \sqrt{x+6}\) starts at the point (-6, 0) and proceeds towards the right, it goes upwards but increases very slowly.

Step 1: Understanding the Basic Graph Transformation

The basic square root function \(y = \sqrt{x}\) starts at the point (0, 0) and then proceeds towards the right, increasing slowly. Adding 6 inside the square root function shifts the graph 6 units leftwards. Thus, instead of starting at (0, 0), our graph will start at (-6, 0).

Step 2: Plotting Points

In order to draw the graph more accurately, it's helpful to plot a few additional points. For instance, when \(x = -4\), then \(y = \sqrt{-4+6} = \sqrt{2}\), and when \(x = 2\), then \(y = \sqrt{2+6} = \sqrt{8}\)

Step 3: Drawing the Graph

Having plotted our points, draw a smooth curve through them starting from (-6, 0) and moving upward and to the right. The graph never crosses the x-axis and y-values only range from 0 to positive infinity because square roots of real numbers are always non-negative.

Key Concepts

Understanding Graph Transformations

When we talk about graph transformations, we mean how the basic shape or direction of a graph changes when certain modifications are applied to the equation. For the square root function, which is written as \(y = \sqrt{x}\), the graph originally starts at the origin (0, 0) and moves rightwards in a gentle curve.
However, if we add or subtract numbers inside the square root (like \(y = \sqrt{x+6}\)), it leads to what's known as a horizontal shift:

• Adding a number shifts the graph to the left.
• Subtracting a number shifts the graph to the right.

In our example, we have \(y = \sqrt{x+6}\). The "+6" causes the entire graph to shift 6 units to the left. That's why it now starts at (-6, 0) instead of (0, 0).
Graph transformations make learning about graphs more intuitive and show how mathematical operations affect graph behavior.

Plotting Points for Accuracy

Plotting points is a crucial step in ensuring that a graph is drawn accurately. It involves selecting a few values for \(x\), calculating the corresponding values of \(y\), and marking these points on a graph.For the function \(y = \sqrt{x+6}\), after shifting it 6 units left, you will start at (-6,0). Choosing some additional points will help you outline the graph's path:

• When \(x = -4\), then \(y = \sqrt{-4 + 6} = \sqrt{2}\).
• When \(x = 2\), then \(y = \sqrt{2 + 6} = \sqrt{8}\).

These calculated points are then plotted to create a general shape for the graph. This ensures that the plot is neither too steep nor too shallow, and follows the correct path as influenced by the transformation.
Plotting gives you a visual guide, making it easier to understand and predict the function's behavior.

The Basic Square Root Function

The basic square root function \(y = \sqrt{x}\) represents the set of all ordered pairs where \(y\) is the square root of \(x\). In its simplest form, this graph starts at the origin (0, 0) since the square root of 0 is 0.Key characteristics of this function include:

• It only takes non-negative values because square roots are non-negative by definition.
• It increases slowly, starting steep and gradually flattening out as \(x\) gets larger.
• It never touches or crosses the x-axis.

Understanding the behavior of \(y = \sqrt{x}\) helps us visualize modifications like transformations. We know that as \(x\) increases, the \(y\) value increases, giving the graph a gentle upward curve. This fundamental understanding is the building block for grasping complex transformations.