Problem 77 - Algebra and Trigonometry Real Mathematics, Real People

Question

Let z-represent a positive real number. Describe how the family of parabolas represented by the given function compares with the graph of \(g(x)=x^{2}\) \(f(x)=(x-z)^{2}\)

Step-by-Step Solution

Answer

The function \( f(x) = (x - z)^2 \) represents a family of parabolas which are identical to the graph of \( g(x) = x^2 \), but translated z units to the right on the x-axis.

Step 1: Understand the base function

The function \( g(x) = x^2 \) is a basic quadratic function, which creates a parabola. The vertex of this parabola is at the origin (0,0) and the parabola opens upwards.

Step 2: Understand the effect of parameter z

In the function \( f(x) = (x - z)^2 \), z represents a horizontal shift of the parabola. If z is a positive real number, it means that our parabola will shift z units to the right compared to our base function \( g(x) = x^2 \). The vertex of the parabola will now be at the point (z, 0).

Step 3: Visualize the comparison

By comparing \( g(x) = x^2 \) and \( f(x) = (x - z)^2 \), it can be seen that the graphs are identical except for a shift along the x-axis. All parabolas will open upwards similarly, as there's no coefficient affecting \( x^2 \) term.

Key Concepts

Horizontal Shift

When we talk about the horizontal shift of a parabola, we refer to moving the entire graph along the x-axis. In our scenario, the function \( f(x) = (x - z)^2 \) exhibits a horizontal shift. Here, \( z \) is a positive real number which means the shift will take place to the right.

To visualize this, compare it with the base function \( g(x) = x^2 \), where the vertex is at the origin, (0, 0). With \( z \), instead of stretching or compressing the graph, we slide it. Each point on the parabola moves \( z \) units to the right, making the new vertex \( (z, 0) \).

Think of it as picking up the graph from its original location and placing it somewhere else along the horizontal direction. The shape and direction remain unchanged but the position is adjusted. Understanding horizontal shift helps you predict where the graph will appear without needing to plot every point.

• The graph moves parallel to the x-axis.
• The shape of the parabola does not change.
• Determined by the sign and value of \( z \), positive shifts right.

Vertex of a Parabola

The vertex of a parabola is a significant point. It represents the maximum or minimum value of the quadratic function, depending on which way the parabola opens. For functions like \( g(x) = x^2 \) and \( f(x) = (x-z)^2 \), the parabola opens upwards.

For the base function \( g(x) = x^2 \), the vertex is at (0, 0). It is the lowest point on the graph, known as the minimum point. When \( z \) is introduced in \( f(x) = (x-z)^2 \), the vertex moves to the point (z, 0). Why? Because \( z \) causes a horizontal shift, as mentioned earlier.

Here's how you can define the vertex:

• It is a turning point, where the graph changes direction.
• For upward-opening parabolas, it's the lowest point.
• When shifted, the vertex shows how far and in what direction the parabola moves.

Understanding the vertex's location aids in graph sketching and identifying changes in position from transformations.

Graph Transformations

Graph transformations involve changing the position or shape of a graph according to mathematical operations. In our exercise, we observed a specific transformation: horizontal shift, which falls under the category of rigid transformations because it alters the position but not the shape.

Consider these transformation types to understand how graphs might be altered:

• Translation: Includes shifts up/down or left/right, like our horizontal shift.
• Reflection: Flips the graph over a line, such as the x-axis or y-axis.
• Dilation: Expands or compresses the graph in a particular direction.

In the case of \( f(x) = (x-z)^2 \), the transformation strictly translates the parabola horizontally, without affecting its shape or the direction it opens. This type of understanding helps in analyzing more complex graph transformations when dealing with additional terms and coefficients in functions.