Problem 29 - Thomas Calculus

Question

Exercises \(25-34\) tell how many units and in what directions the graphs of the given equations are shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation. $$ y=\sqrt{x} \text { Left } 0.81 $$

Step-by-Step Solution

Answer

Shifted equation is \( y = \sqrt{x + 0.81} \).

Step 1: Identify the Basic Function

The original function is given as \( y = \sqrt{x} \). This represents the parent function, a basic square root function.

Step 2: Analyze the Transformation

The problem states a horizontal shift to the left by 0.81 units. A horizontal shift to the left involves changing the expression inside the square root from \( x \) to \( x + ext{{shift amount}} \). Thus, the transformed equation is \( y = \sqrt{x + 0.81} \).

Step 3: Write the Transformed Equation

After applying the horizontal shift to the left, the shifted equation is \( y = \sqrt{x + 0.81} \). This equation represents the graph of the original function moved 0.81 units to the left.

Step 4: Sketch the Graphs

To sketch the graphs, plot the original function \( y = \sqrt{x} \) with its characteristic shape that starts at the origin (0,0) and curves upwards and to the right. Then, sketch the shifted function \( y = \sqrt{x + 0.81} \), which will start slightly left of the origin (-0.81, 0) and follow the same curvature. Clearly label each graph with its respective equation.

Key Concepts

Horizontal Shift

A horizontal shift is a transformation that moves a graph left or right along the x-axis.
It changes the position of a graph but not its shape. Here's how it works with the square root function. To perform a horizontal shift, you modify the inside of the function's formula. For example:

• If you want to shift the graph of the function to the left, you add a number to the variable inside the function. For instance, to shift left by 0.81 units, you change from \( y = \sqrt{x} \) to \( y = \sqrt{x + 0.81} \).


• If shifting to the right, you subtract from the variable: \( y = \sqrt{x} \) would become \( y = \sqrt{x - 0.81} \).

Understanding horizontal shifts is crucial for sketching transformed graphs accurately, as it tells you exactly how the graph's position changes relative to its original alignment.

Graph Sketching

Graph sketching is a valuable skill that helps visualize mathematical functions.
It involves plotting the graph of the function to see how it behaves over a range of values. When sketching the graph of a function like \( y = \sqrt{x} \), you start by identifying key features:

• The initial point, at (0,0), is where the graph begins.


• The shape of the graph, which is a curved line going upwards and to the right.

After finding these features, sketching the graph is straightforward:
Simply plot the initial point and extend the curve following the expected behavior.
If you've applied a horizontal shift, adjust the starting point according to the shift.
For \( y = \sqrt{x + 0.81} \), the starting point shifts to (-0.81, 0), while the graph's shape remains unchanged.
Always label your graphs to avoid confusion, which helps in clearly distinguishing between different transformations.

Square Root Function

The square root function is one of the fundamental functions in mathematics.
It is expressed as \( y = \sqrt{x} \) and has distinct characteristics that are useful in many areas. Key points about the square root function include:

• The domain, which consists of non-negative numbers (\( x \geq 0 \)), since you can't take the square root of a negative number in real numbers.


• The range, which includes non-negative numbers as well (\( y \geq 0 \)), due to how the square root operation works.


• The basic "hook" shape of its graph, starting at the origin (0,0) and extending upwards to the right.

Understanding these features helps you apply transformations effectively.
Whether shifting the graph horizontally, vertically, or both, the function's core properties remain essential for making accurate sketches and interpretations.