how do i know how much to dilate a parabola by (how to know how much more narrow or fatter i should make the parabola

To determine how much to dilate a parabola, you should look at the **coefficient of the \(x^2\) term** in the quadratic equation that defines the parabola. The general form of a quadratic equation is:

\[
y = ax^2 + bx + c
\]

Here’s how the coefficient \(a\) affects the width (or "fatness") of the parabola:

1. **If \( |a| > 1 \)**: The parabola is **narrower** than the standard parabola \(y = x^2\). The greater the value of \( |a| \), the narrower the parabola.

2. **If \( |a| < 1 \)**: The parabola is **wider** or "fatter" than the standard parabola \(y = x^2\). The smaller the value of \( |a| \) (closer to 0 but not 0), the wider the parabola.

3. **If \( a = 1 \)**: The parabola has the same width as the standard parabola \(y = x^2\).

4. **If \( a = -1 \)**: The parabola is the same width as the standard parabola but opens **downward**.

5. **If \( a < 0 \)**: The parabola opens downward, and the magnitude of \(a\) still controls the width similarly as described above.

### How to Adjust the Width:
- **To make the parabola narrower**: Increase the magnitude of \(a\) (e.g., change \(a\) from 1 to 3).
- **To make the parabola wider**: Decrease the magnitude of \(a\) (e.g., change \(a\) from 1 to 0.5).

The value of \(a\) directly scales the parabola vertically, which in turn affects how wide or narrow it appears.

Would you like more details or have any questions?

### Related Questions:
1. How does the vertex form of a parabola relate to its dilation?
2. What effect does changing the sign of \(a\) have on the parabola’s orientation?
3. How does the dilation of a parabola affect its focal length?
4. Can the dilation of a parabola change the position of its vertex?
5. How does the dilation of a parabola impact its intercepts with the x-axis?

### Tip:
When graphing parabolas, plotting a few key points before and after dilation can help visualize how the parabola's width changes.

Learn how to determine the amount to dilate a parabola and adjust its width. This guide covers the effects of the coefficient of the x^2 term on the parabola's width, with practical examples and adjustments. Suitable for high school students studying quadratic equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation