Question

Explain in detail the role of 𝑎, ℎ, and 𝑘 in the equation 𝑦 = 𝑎(𝑥 − ℎ)^2 + 𝑘. You

may use words, numerical examples, and sketches to support your explanation.

Answer 1

[AI-generated answer — Bot]

In the equation 𝑦 = 𝑎(𝑥 − ℎ)^2 + 𝑘, 𝑎, ℎ, and 𝑘 are constants that affect the shape, position, and orientation of the parabolic curve.

- 𝑎: The parameter 𝑎 determines the shape of the parabola. Specifically, it determines whether the parabola opens upwards (𝑎 > 0) or downwards (𝑎 < 0) and how wide or narrow the curve is. Larger values of 𝑎 lead to steeper and skinnier curves, while smaller values of 𝑎 lead to flatter and wider curves. For example, if 𝑎 = 1, the resulting parabola would look like a "U", whereas if 𝑎 = -1, the parabola would look like an upside-down "U".

- ℎ: The value of ℎ corresponds to the x-coordinate of the vertex of the parabolic curve. The vertex represents the highest or lowest point on the curve, depending on whether the parabola opens upwards or downwards. If ℎ > 0, the vertex is shifted to the right, and if ℎ < 0, the vertex is shifted to the left. For example, if ℎ = 2, the vertex would be located at x = 2.

- 𝑘: The parameter 𝑘 determines the height of the parabola, or where the curve intersects with the y-axis. If 𝑘 > 0, the parabola is shifted upwards from its original position, and if 𝑘 < 0, the parabola is shifted downwards. For example, if 𝑘 = 3, the parabola would intersect the y-axis at y = 3.

To illustrate the effect of these parameters visually, consider the following examples:

Example 1: 𝑦 = -2(𝑥 - 3)^2 + 5

Here, 𝑎 = -2, ℎ = 3, and 𝑘 = 5. The negative value of 𝑎 means that the parabola opens downwards, and the absolute value of 𝑎 = 2 indicates that the curve is relatively narrow. The vertex is located at (3, 5), and since 𝑘 > 0, the parabola is shifted upwards by 5 units. The resulting graph looks like:


Example 2: 𝑦 = 0.5(𝑥 + 1)^2 - 2

Here, 𝑎 = 0.5, ℎ = -1, and 𝑘 = -2. The positive value of 𝑎 indicates that the parabola opens upwards, and the small value of 𝑎 = 0.5 indicates that the curve is relatively wide. The vertex is located at (-1, -2), and since 𝑘 < 0, the parabola is shifted downwards by 2 units. The resulting graph looks like:


Answer 2

[AI-generated answer — Step-by-Step Bot]

In the equation 𝑦 = 𝑎(𝑥 − ℎ)^2 + 𝑘, 𝑎, ℎ, and 𝑘 are all constants that affect the shape and position of the graph of the equation.

1. 𝑎 (the coefficient of the quadratic term):
- The value of 𝑎 determines the amount of "stretch" or "compression" of the parabola.
- When 𝑎 > 0, the parabola opens upward. The larger the value of 𝑎, the narrower the parabola becomes.
- When 𝑎 < 0, the parabola opens downward. The magnitude of 𝑎 determines the width of the parabola.
- If 𝑎 = 1, the parabola keeps its standard shape.
- If 𝑎 > 1, the parabola is stretched vertically.
- If 0 < 𝑎 < 1, the parabola is compressed vertically.

Example: Let's take 𝑦 = 2(𝑥 - 1)^2 + 3 as an example. Since 𝑎 = 2, the parabola opens upward and is relatively narrow.

2. ℎ (the x-coordinate of the vertex):
- ℎ determines the horizontal shift of the parabola.
- When ℎ > 0, the parabola shifts ℎ units to the right.
- When ℎ < 0, the parabola shifts ℎ units to the left.

Example: Continuing from the previous example, if we change ℎ from 1 to 3 in 𝑦 = 2(𝑥 - 1)^2 + 3, the vertex of the parabola will shift two units to the right.

3. 𝑘 (the y-coordinate of the vertex):
- 𝑘 determines the vertical shift of the parabola.
- When 𝑘 > 0, the parabola shifts 𝑘 units upward.
- When 𝑘 < 0, the parabola shifts 𝑘 units downward.

Example: Let's take 𝑦 = 2(𝑥 - 1)^2 + 3 as an example again. Since 𝑘 = 3, the vertex of the parabola is located at (1, 3), which means it shifts three units upward.

Overall, changing the values of 𝑎, ℎ, and 𝑘 in the equation 𝑦 = 𝑎(𝑥 − ℎ)^2 + 𝑘 affects the shape, position, and orientation of the parabola.