Question

Graph the function \(f\) by starting with the graph of \(y=x^{2}\) and using transformations. \(f(x)=\frac{1}{4} x^{2}\)

Short answer

\(f(x) = \frac{1}{4}x^2\) is a vertically compressed parabola compared to \(y = x^2\).

Step-by-step solution

- Identify the base function

The base function given is the standard quadratic function defined by the equation \(y = x^2\). This function produces a parabola that opens upwards with its vertex at the origin (0,0).

- Understand the transformation

The given function is \(f(x) = \frac{1}{4}x^2\). This represents a vertical compression of the base function \(y = x^2\). The factor \(\frac{1}{4}\) indicates that every y-value of the base function will be multiplied by \(\frac{1}{4}\), making the parabola wider.

- Apply the transformation to key points

To graph \(f(x)=\frac{1}{4} x^{2}\), take key points from the base function \(y=x^2\) like (0,0), (1,1), (-1,1), (2,4), and (-2,4). Multiply the y-values of these points by \(\frac{1}{4}\):

- Calculate transformed points

The new points after the transformation will be:- \((0, 0) \rightarrow (0, 0)\)- \((1, 1) \rightarrow (1, \frac{1}{4})\)- \((-1, 1) \rightarrow (-1, \frac{1}{4})\)- \((2, 4) \rightarrow (2, 1)\)- \((-2, 4) \rightarrow (-2, 1)\)

- Plot and draw the graph

Plot the transformed points onto a coordinate system: (0,0), (1, \(\frac{1}{4}\)), (-1, \(\frac{1}{4}\)), (2,1), and (-2,1). Draw a smooth curve through these points to complete the graph of the function \( f(x) = \frac{1}{4}x^2 \). The resulting graph will be a wider parabola than the original graph of \( y = x^2 \).

Key concepts

Quadratic Transformation

Quadratic transformations involve changes to the standard quadratic function, which is represented by the equation \( y = x^2 \). These transformations can alter the graph's position and shape. The basic quadratic function produces a parabola that opens upwards, centered at the origin (0,0). If we modify this original function using various alterations like stretching, compressing, translating, or reflecting, we perform a quadratic transformation.
In our example, the function provided is \( f(x) = \frac{1}{4}x^2 \). This transformation does not move the graph horizontally or vertically, but it does affect its shape. Specifically, it introduces a vertical compression. This change makes the parabola wider compared to the base function \( y = x^2 \).

Understanding these transformations is crucial for graphing quadratic functions because it helps to predict the changes in the graph's appearance quickly.

Vertical Compression

Vertical compression is a type of transformation that affects the width of a parabola. In the context of our exercise, the given function \( f(x) = \frac{1}{4}x^2 \) compresses the original quadratic function \( y = x^2 \) vertically. This means every y-value of the function is reduced by multiplying by \( \frac{1}{4} \).
When applying vertical compression:

• \

Key Points

Key points are crucial for accurately drawing the graph of a quadratic function. They help us understand the changes brought by transformations and ensure that the graph is plotted correctly. To graph \( f(x) = \frac{1}{4}x^2 \), we start from the key points of the original function \( y = x^2 \). - Original Key Points:

• (0, 0)
• (1, 1)
• (-1, 1)
• (2, 4)
• (-2, 4)

Applying the transformation \( f(x)= \frac{1}{4}x^2 \) involves multiplying the y-values by \( \frac{1}{4} \): - Transformed Key Points:

• (0, 0) to (0, 0)
• (1, 1) to (1, \( \frac{1}{4} \))
• (-1, 1) to (-1, \( \frac{1}{4} \))
• (2, 4) to (2, 1)
• (-2, 4) to (-2, 1)

Plotting these transformed points allows us to see that the new parabola, although wider than the original, still maintains its fundamental shape and opens upwards. By connecting these points smoothly, we complete the graph of \( f(x) = \frac{1}{4}x^2 \).