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

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.