Problem 33 - Algebra and Trigonometry

Question

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=\frac{1}{4} x^{2} $$

Step-by-Step Solution

Answer

The graph of \(y = \frac{1}{4} x^2\) is a vertically compressed parabola centered at the origin.

Step 1: Identify the Standard Form

The given function is \(y = \frac{1}{4} x^2\). The standard form of this function is the quadratic function \(y = x^2\), which is a parabola that opens upwards with vertex at the origin (0,0).

Step 2: Apply Vertical Scaling

The coefficient \(\frac{1}{4}\) in front of \(x^2\) indicates a vertical scaling of the standard parabola. This scaling compresses the parabola vertically, making it wider than the standard \(y = x^2\).

Step 3: Sketch the Transformed Graph

To sketch \(y = \frac{1}{4} x^2\), start with the graph of \(y = x^2\), and apply the vertical compression. The graph will still have its vertex at (0,0) but will be wider than the standard parabola due to the factor \(\frac{1}{4}\).

Key Concepts

Parabola

A parabola is a U-shaped curve that is the graphical representation of a quadratic function. It has a distinct symmetry and can open upwards or downwards. When dealing with the function \( y = x^2 \), this parabola opens upwards and its vertex is located at the origin, which is the point (0,0). Understanding this standard shape is crucial.
- The vertex is the highest or lowest point, depending on the parabola's direction.- The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves.- For the parabola described by \( y = x^2 \), the axis of symmetry is the y-axis.
In practical application, knowing this baseline function helps in identifying how the parabola transforms with various function adjustments, including shifts, scaling, and reflections.

Vertical Scaling

Vertical scaling involves altering the width of the parabola by multiplying the quadratic term by a coefficient. In \( y = \frac{1}{4}x^2 \), the coefficient \( \frac{1}{4} \) impacts the parabola’s shape. Here's how it works:

• A coefficient greater than 1 stretches the parabola, making it "taller" and "narrower".
• A coefficient between 0 and 1, like \( \frac{1}{4} \), compresses the parabola, resulting in a "shorter" and "wider" graph.
• A negative coefficient would invert the parabola, flipping it upside down.

For \( y = \frac{1}{4}x^2 \), the vertical compression makes the parabola wider compared to the standard \( y = x^2 \). This transformation maintains the vertex at (0,0), changing only the spread of the arms of the parabola.

Function Transformations

Function transformations entail adjusting the graph of a function to produce a new graph by shifting, reflecting, or scaling. While the equation \( y = \frac{1}{4}x^2 \) primarily involves scaling, understanding other transformations is beneficial.
- **Translations** move the graph horizontally or vertically without altering its shape. - Adding or subtracting a constant inside the function shifts it horizontally. - Adding or subtracting a constant outside the function shifts it vertically.- **Reflections** flip the graph across an axis. For quadratic functions, a negative sign before \( x^2 \) reflects it over the x-axis.- **Scaling** changes the width or height, as seen in the function provided, where the compression widens the parabola.
Recognizing these aspects allows one to analyze and sketch transformed functions by adapting the standard form through these intuitive adjustments.