Problem 39 - Beginning and Intermediate Algebra

Question

Use the transformation techniques to graph each of the following functions. $$y=(x-3)^{2}+1$$

Step-by-Step Solution

Answer

The graph of the function \(y = (x-3)^2 + 1\) is obtained by applying a horizontal shift of 3 units to the right and a vertical shift of 1 unit upwards to the parent function \(y=x^2\). The final graph is a parabola with a vertex at (3, 1) and opens upwards.

Step 1: Identify the parent function and transformations

We have the function \( y = (x-3)^2 + 1 \), and we can recognize the parent function as \( y = x^2 \). The given function shows that a horizontal shift and a vertical shift have been applied to the parent function. The horizontal shift is by adding 3 units inside the parenthesis (i.e., \(x-3\)), which indicates that the graph will move to the right by 3 units. The vertical shift is by adding 1 outside the square, which indicates that the graph will move up by 1 unit.

Step 2: Graph the parent function and apply horizontal shift

First, draw the graph of the parent function \( y = x^2 \) with a vertex at (0, 0) and opening upwards. Apply the horizontal shift by moving the vertex to the right by 3 units. After this transformation, the vertex's position will be at (3, 0) and the shape of the graph remains the same.

Step 3: Apply the vertical shift

Now, apply the vertical shift to the function by moving the graph up by 1 unit. The vertex will move from (3, 0) to (3, 1). The shape of the graph remains the same after this transformation.

Step 4: Draw the final graph

Based on the transformations applied in Steps 2 and 3, the final graph of the function \( y = (x-3)^2 + 1 \) is a parabola with a vertex at (3, 1) that opens upwards. Keep the same shape as the parent function \(y = x^2\), as no stretching or shrinking transformations were applied. Consequently, the graph of the function \(y = (x-3)^2 + 1\) is obtained by applying a horizontal shift of 3 units to the right and a vertical shift of 1 unit upwards to the parent function \(y=x^2\).

Key Concepts

Transformation Techniques

When graphing quadratic functions, transformation techniques are crucial as they help us modify the appearance of the graph in various ways. They enable us to adjust the function’s position, orientation, and shape on a coordinate plane.
Common transformation techniques used for quadratics include:

• Horizontal shifts: Move the graph left or right.
• Vertical shifts: Move the graph up or down.
• Reflections: Flip the graph over a specific axis.
• Stretching/Shrinking: Change the width of the graph.

Understanding how these transformations appear in the equation of a function will help you graph the function accurately and recognize the new position of key points like the vertex.

Parent Function

The parent function provides a basic shape or 'template' from which other, more complex, functions are derived. For quadratic functions, the simplest form usually is \( y = x^2 \).
Key characteristics of the parent function include:

• Shape: A parabola that's centrally symmetrical around the y-axis.
• Vertex: Located at the origin (0,0).
• Direction: Opens upwards.

By understanding the parent function, you can easily identify how other quadratic functions differ based on transformations like shifts, stretches, or reflections applied.

Horizontal Shift

A horizontal shift changes the graph's position along the x-axis. This occurs when you add or subtract a constant from the \( x \) variable within a function.
For the function \( y = (x-3)^2 + 1 \), the expression \( x-3 \) indicates a shift. Specifically, it shifts the graph of the parent function \( y = x^2 \) 3 units to the right.
Here’s how you determine the shift direction:

• Inside the parenthesis: \( x-h \)
• "+" value moves left
• "-" value moves right

Horizontal shifts do not alter the graph's shape itself, just its location on the x-axis.

Vertical Shift

A vertical shift involves moving the graph up or down along the y-axis. This happens when a constant is added or subtracted from the whole function.
In the equation \( y = (x-3)^2 + 1 \), the "+1" outside the square represents a vertical shift. This specific shift moves the entire graph up by 1 unit.
Here's how vertical shifts are understood:

• + k indicates an upward shift by \( k \) units.
• - k indicates a downward shift by \( k \) units.

Like horizontal shifts, vertical shifts do not affect the shape of the graph but alter its position on the y-axis. Understanding both vertical and horizontal shifts helps in accurately sketching the modified graphs with the right vertex and orientation.