Problem 72 - Algebra for College Students

Question

Graph \(y=x^{2}, y=(x-3)^{2},\) and \(y=(x+3)^{2}\) on the same coordinate system. How does the graph of \(y=(x-h)^{2}\) compare to the graph of \(y=x^{2} ?\)

Step-by-Step Solution

Answer

The graph of \(y = (x-h)^{2}\) is a horizontal shift of the graph \(y = x^{2}\) by \(h\) units.

Step 1 - Graph the parent function

Graph the function \(y = x^{2}\) on the coordinate system. This is a standard parabola that opens upwards with its vertex at the origin (0, 0).

Step 2 - Graph the shifted function to the right

Graph the function \(y = (x-3)^{2}\). This parabola is the same shape as \(y = x^{2}\) but shifted to the right by 3 units. Its vertex is at (3, 0).

Step 3 - Graph the shifted function to the left

Graph the function \(y = (x+3)^{2}\). This parabola is the same shape as \(y = x^{2}\) but shifted to the left by 3 units. Its vertex is at (-3, 0).

Step 4 - Compare the graphs

Observe that \(y = x^{2}\) is centered at the origin, \(y = (x-3)^{2}\) is shifted right by 3 units, and \(y = (x+3)^{2}\) is shifted left by 3 units. In general, the graph of \(y = (x-h)^{2}\) is the same as \(y = x^{2}\) but shifted horizontally by \(h\) units.

Key Concepts

Horizontal Shifts

When graphing parabolas, horizontal shifts can be quite simple once you understand the basics. The parent function, which is the simplest form of a function family, for a parabola is given by the equation: \(y = x^2\).
The parabola of this equation is centered at the origin with its vertex at (0, 0).
When you modify the equation to \(y = (x - h)^2\), the entire parabola shifts horizontally.
Specifically:

• If you subtract a positive value, like in \(y = (x - 3)^2\), the graph shifts to the right by that value (3 units to the right in this case).
• If you add a positive value, like in \(y = (x + 3)^2\), the graph shifts to the left by that value (3 units to the left in this case).

It is important to remember that these shifts only affect the position of the vertex horizontally and do not alter the shape of the parabola.
The shape and orientation (opening upwards) remain the same regardless of the shift.

Vertex of a Parabola

Understanding the vertex of a parabola is crucial in graphing. The vertex is the highest or lowest point on the graph of a quadratic function, depending on the parabola’s orientation.
In the parent function \(y = x^2\), the vertex is at the origin (0, 0). When dealing with the equation \(y = (x - h)^2\), the vertex shifts horizontally based on the value of \(h\). For example:

• In \(y = (x - 3)^2\), the vertex moves to (3, 0).
• In \(y = (x + 3)^2\), the vertex shifts to (-3, 0).

To find the vertex quickly:

• Identify the value of \(h\) in \( (x - h)^2\).
• The vertex will be at (\(h\), 0).

This method helps you quickly predict and plot the vertex on the graph, then you can sketch the rest of the parabola knowing its shape remains unchanged.

Parent Function

The term 'parent function' refers to the simplest version of a function type. For parabolas, the parent function is \(y = x^2\). This equation represents a set standard from which all other quadratic equations derive.
The graph of this parent function is a symmetrical parabola that opens upward and its vertex is at the origin (0, 0).
Utilizing the parent function allows us to understand transformations like horizontal shifts and vertical shifts easily.
For example, modifying the parent function to \(y = (x - h)^2\) helps us visualize horizontal translations without altering the parabola’s shape.
Besides horizontal shifts, other transformations can include:

• Vertical shifts by changing the equation to \(y = x^2 + k\).
• Stretching or compressing vertically by modifying it to \(y = a \times x^2\).

Understanding the parent function provides a foundation for graphing more complex quadratic equations efficiently.