Problem 9 - Precalculus : Building Concepts and Connections

Question

Complete them to review topics relevant to the remaining exercises. The graph of \(f(x)=x^{2}+3\) is the graph of \(y=x^{2}\) shifted __________ 3 units.

Step-by-Step Solution

Answer

Upwards

Step 1: Understanding the parent function

The parent function here is \(f(x) = x^{2}\), which is a parabola that opens upwards with the vertex at the origin (0,0).

Step 2: Recognizing the transformation

In the function \(f(x) = x^{2} + 3\), the '+3' shifts the graph upwards by 3 units.

Step 3: Final Answer

Therefore, the graph of \(f(x) = x^{2} + 3\) is the graph of \(y = x^{2}\) shifted upwards by 3 units.

Key Concepts

Graph Transformations

Graph transformations are changes made to the parent function to produce a new graph. The parent function \(f(x) = x^2\) is a basic quadratic function, which creates a standard parabola shape. Transformations can include shifting, stretching, or reflecting the graph.

Simplifying the concept, when you apply transformations to a graph, you modify its position or shape on the coordinate plane. In our case, the transformation is applied to the parabola described by the function \(f(x) = x^2 + 3\), a common type of transformation known as a vertical shift.

This type of transformation is beneficial in analyzing how different types of manipulation affect the graph, whether it be vertical or horizontal shifts, reflections across axes, or changes in how narrow or wide the parabola appears.

Parabola Characteristics

A parabola is a U-shaped graph that represents a quadratic function. Commonly, parabolas can open upwards or downwards depending on the direction of the 'U'. The vertex of the parabola, which is the highest or lowest point of the graph, changes its position based on transformations.

Important characteristics of a parabola include:

• Vertex: The point where the parabola reaches its maximum or minimum value. For the parent function \(f(x) = x^2\), the vertex is at (0,0).
• Axis of Symmetry: A vertical line that divides the parabola into two equal mirror images. In our parent function, it is the y-axis or equation \(x = 0\).
• Direction: Determines whether the parabola opens up or downwards. When the coefficient of the \(x^2\) term is positive, it opens upwards.
• Focus and Directrix: Terms that are mathematically utilized to define the parabola, though less often used in basic transformations.

Parabolas have these consistent features, but understanding how they adjust with transformations is crucial for graph interpretation.

Vertical Shift

A vertical shift is an alteration where the graph moves up or down along the y-axis. If we look at the transformation from \(f(x) = x^2\) to \(f(x) = x^2 + 3\), it involves a vertical shift.

Here's how to recognize a vertical shift in quadratic functions:

• Vertical Shift Up: When a constant is added to the function, as with \(f(x) = x^2 + 3\), it shifts the entire graph upwards by that constant value. The shift up by 3 means all points on the graph move 3 units higher.
• Vertical Shift Down: When a constant is subtracted, the graph shifts downwards, moving all points on the parabola lower by that constant value.

Vertical shifts do not affect the shape of the graph or its opening direction; they simply transfer it up or down, changing the location of its vertex. For \(f(x) = x^2 + 3\), this results in a vertex moving from (0, 0) to (0, 3), retaining all other characteristics of the original parabola.