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

The equation 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 represents a quadratic function where 𝑎, ℎ, and 𝑘 play different roles in determining the shape, position, and orientation of the graph.

𝑎 represents the coefficient of the quadratic term and determines the direction and degree of the opening of the parabola. If 𝑎 is positive, the parabola opens upward, and if 𝑎 is negative, the parabola opens downward. The larger the absolute value of 𝑎, the steeper the slope of the parabola, and the narrower the width. For example, the graphs of 𝑦 = 2(𝑥 − 1)2 + 1 and 𝑦 = −2(𝑥 − 1)2 + 1 are shown below, where ℎ is fixed at 1 and 𝑘 is fixed at 1. The red graph has a minimum point at (1, 1) and opens upward, while the blue graph has a maximum point at (1, 1) and opens downward.



ℎ represents the horizontal shift or translation of the parabola on the x-axis. The value of ℎ indicates the x-coordinate of the vertex, which is the highest or lowest point of the parabola depending on the value of 𝑎. If ℎ is positive, the parabola shifts to the right, and if ℎ is negative, the parabola shifts to the left. For example, the graphs of 𝑦 = 2(𝑥 − 1)2 + 1 and 𝑦 = 2(𝑥 + 3)2 + 1 are shown below, where 𝑎 and 𝑘 are fixed at 2 and 1, respectively. The red graph has a vertex at (1, 1), while the blue graph has a vertex at (−3, 1).



𝑘 represents the vertical shift or translation of the parabola on the y-axis. The value of 𝑘 indicates the y-coordinate of the vertex, which is added or subtracted from the square term 𝑎(𝑥 − ℎ)2. If 𝑘 is positive, the parabola shifts upward, and if 𝑘 is negative, the parabola shifts downward. For example, the graphs of 𝑦 = 2(𝑥 − 1)2 + 1 and 𝑦 = 2(𝑥 − 1)2 − 2 are shown below, where 𝑎 is fixed at 2 and ℎ is fixed at 1. The red graph has a vertex at (1, 1), while the blue graph has a vertex at (1, −2).



In summary, the role of 𝑎, ℎ, and 𝑘 in the equation 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 are:

- 𝑎 determines the direction and degree of the opening of the parabola.
- ℎ represents the horizontal shift or translation of the parabola on the x-axis.
- 𝑘 represents the vertical shift or translation of the parabola on the y-axis.