Problem 33 - A Graphical Approach to Precalculus with Limits

Question

Give the equation of each function whose graph is described. The graph of \(y=x^{2}\) is vertically shrunk by applying a factor of \(\frac{1}{2},\) and the resulting graph is shifted 7 units downward.

Step-by-Step Solution

Answer

The equation is \( y = \frac{1}{2}x^2 - 7 \).

Step 1: Analyze the Original Function

The original function given is \( y = x^2 \). This is a standard parabola opening upwards.

Step 2: Apply Vertical Shrink

To apply a vertical shrink by a factor of \( \frac{1}{2} \), multiply the entire function by \( \frac{1}{2} \). The modified function becomes \( y = \frac{1}{2}x^2 \).

Step 3: Shift the Graph Downward

To shift the graph downward by 7 units, subtract 7 from the entire function. This transforms the function to \( y = \frac{1}{2}x^2 - 7 \).

Step 4: Combine Transformations

The combined transformations give the final equation of the function as \( y = \frac{1}{2}x^2 - 7 \).

Key Concepts

Understanding Quadratic Functions

Quadratic functions are fundamental parts of algebra and are essential for understanding various mathematical concepts. These functions are typically in the form \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. The graph of a quadratic function is a parabola, which can open upwards or downwards, depending on the sign of \( a \). If \( a > 0 \), the parabola opens upwards, resembling a cup or a "U" shape.

Key features of quadratic functions include:

• Vertex: The highest or lowest point on the graph. If the parabola opens upwards, the vertex is the minimum point; if it opens downwards, the vertex is the maximum point.
• Axis of Symmetry: A vertical line that passes through the vertex and divides the parabola into two symmetrical parts. For the basic function \( y = x^2 \), the axis of symmetry is the y-axis.
• Roots: Also known as x-intercepts or zeros, these are the points where the graph intersects the x-axis.

Quadratic functions serve as a building block for more complex algebraic and geometric equations, making them crucial in various real-world applications.

Applying a Vertical Shrink

A vertical shrink is a transformation applied to a function that compresses the graph towards the x-axis. It involves multiplying the function by a factor between 0 and 1. For example, applying a vertical shrink to \( y = x^2 \) by a factor of \( \frac{1}{2} \) results in the function \( y = \frac{1}{2}x^2 \).

This transformation affects how "steep" or "flat" the graph appears:

• Compression: The graph becomes flatter, indicating that the shape is compressed vertically, but retains its general direction. In our example, the original parabola becomes less steep, making it appear wider.
• Stretch (Counter Concept): In contrast, a vertical stretch would involve multiplying by a factor greater than 1, making the graph steeper.

Vertical shrinks are particularly useful for altering the graph's appearance without changing its basic shape or other characteristics like symmetry.

Explaining the Vertical Shift

A vertical shift moves the entire graph of a function up or down without altering its shape or orientation. For a quadratic function like \( y = \frac{1}{2}x^2 \), applying a vertical shift can dramatically change where the graph is positioned relative to the Cartesian coordinate system.

In the given exercise, the graph is shifted 7 units downward:

• Downward Shift: To achieve this, we subtract 7 from the function. The transformation results in \( y = \frac{1}{2}x^2 - 7 \), which moves the entire graph 7 units down.
• Upward Shift (Counter Concept): Conversely, adding a positive constant to the function's equation would shift the graph upwards.

This transformation is particularly useful for positioning the graph at a specific vertical locale, which can be critical in applications where the function needs to be aligned with other elements or reference points in space.