Question

Explain 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]

The equation 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 is in standard form and represents a parabola.

- The value of 𝑘 is the 𝑦-coordinate of the vertex of the parabola.
- The value of ℎ is the 𝑥-coordinate of the vertex of the parabola.
- The value of 𝑎 determines the shape, direction, and steepness of the parabola.

To illustrate this, consider the following parabolas:

- If 𝑎 > 0, the parabola opens upwards (like a smiley face), and its vertex is the lowest point. For example, 𝑦 = 2(𝑥 − 3)2 + 1 has a vertex of (3,1), and the parabola looks like this:

- If 𝑎 < 0, the parabola opens downwards (like a frown), and its vertex is the highest point. For example, 𝑦 = −3(𝑥 + 2)2 + 4 has a vertex of (−2,4), and the parabola looks like this:

- If 𝑎 = 0, the equation is that of a horizontal line with a 𝑦-intercept of 𝑘. For example, 𝑦 = 7 has a horizontal line with a 𝑦-intercept of 7.

In summary, 𝑎 controls the steepness of the parabola, ℎ shifts the parabola left or right, and 𝑘 raises or lowers the parabola.

Answer 2

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

In the equation 𝑦 = 𝑎(𝑥 − ℎ)² + 𝑘, the variables 𝑎, ℎ, and 𝑘 have specific roles and influence the shape, position, and the direction of the graph of the quadratic function.

1. Role of 𝑎:
- The variable 𝑎 is the coefficient of the quadratic term in the equation.
- It determines the stretch or compression of the parabola along the vertical axis.
- When 𝑎 > 0, the parabola opens upwards and is stretched vertically. The larger the value of 𝑎, the narrower the parabola becomes. For example, if 𝑎 = 2, the graph will be narrower than if 𝑎 = 1.
- When 𝑎 < 0, the parabola opens downwards and is compressed vertically. The absolute value of 𝑎 determines the steepness of the parabola. For example, if 𝑎 = -2, the graph will be steeper than if 𝑎 = -1.

2. Role of ℎ:
- The variable ℎ represents the horizontal shift of the vertex of the parabola.
- The vertex of the parabola is located at the point (ℎ, 𝑘). If ℎ > 0, the vertex will be shifted to the right; if ℎ < 0, the vertex will be shifted to the left. The magnitude of ℎ determines the amount of the horizontal shift.
- For example, if ℎ = 3, the vertex will be shifted 3 units to the right from the origin (0, 0). If ℎ = -2, the vertex will be shifted 2 units to the left.

3. Role of 𝑘:
- The variable 𝑘 represents the vertical shift of the vertex of the parabola.
- The vertex is located at the point (ℎ, 𝑘). If 𝑘 > 0, the vertex will be shifted upward; if 𝑘 < 0, the vertex will be shifted downward. The magnitude of 𝑘 determines the amount of the vertical shift.
- For example, if 𝑘 = 5, the vertex will be shifted 5 units upward from the origin (0, 0). If 𝑘 = -3, the vertex will be shifted 3 units downward.

Here is a numerical example to illustrate the role of 𝑎, ℎ, and 𝑘 using the equation 𝑦 = 2(𝑥 − 3)² + 5:
- 𝑎 = 2, which means the parabola opens upwards and is stretched vertically.
- ℎ = 3, which means the vertex is shifted 3 units to the right.
- 𝑘 = 5, which means the vertex is shifted 5 units upward.

Sketching the graph of this equation will help visualize the role of 𝑎, ℎ, and 𝑘 accurately.