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]

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.

Answer 2

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

In the equation 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘, 𝑎, ℎ, and 𝑘 are constants that determine the shape, position, and orientation of the parabola represented by the equation.

1. The value of 𝑎:
The coefficient 𝑎 determines the stretch or compression of the parabola along the vertical axis. A positive value of 𝑎 makes the parabola open upwards, while a negative value of 𝑎 makes it open downwards. The magnitude of 𝑎 affects the steepness of the opening of the parabola. A larger magnitude value of 𝑎 results in a steeper parabola, while a smaller magnitude value of 𝑎 leads to a flatter parabola.

Example: Let's take 𝑎 = 2, and see the effect on the graph of the equation.
𝑦 = 2(𝑥 − ℎ)2 + 𝑘

If ℎ = 0 and 𝑘 = 0, the equation becomes 𝑦 = 2𝑥^2.
Using a table of values, we can plot some points to create a graph.

x | y
---------------
-2 | 8
-1 | 2
0 | 0
1 | 2
2 | 8

Graph of the equation 𝑦 = 2𝑥^2:
.
.
.
- - - - - - - - - - - - -

As you can see, the parabola opens upwards and is steeper compared to the graph of a standard parabola.

2. The value of ℎ:
The constant ℎ represents the horizontal shift (or translation) of the vertex of the parabola from the standard position (0,0) on the coordinate plane. A positive value of ℎ shifts the parabola horizontally to the right, while a negative value of ℎ shifts it to the left.

Example: Let's take ℎ = 3, and keep 𝑎 = 1 and 𝑘 = 0 for simplicity.
𝑦 = (𝑥 − 3)^2

Using a table of values, we can plot some points to create a graph.

x | y
---------------
1 | 4
2 | 1
3 | 0
4 | 1
5 | 4

Graph of the equation 𝑦 = (𝑥 − 3)^2:
- - - - - - - - - - - - -
.
.
.
.

As you can see, the parabola is shifted 3 units to the right compared to the graph of a standard parabola.

3. The value of 𝑘:
The constant 𝑘 represents the vertical shift (or translation) of the vertex of the parabola from the standard position (0,0) on the coordinate plane. A positive value of 𝑘 shifts the parabola vertically upwards, while a negative value of 𝑘 shifts it downwards.

Example: Let's take 𝑘 = -2, and keep 𝑎 = 1 and ℎ = 0 for simplicity.
𝑦 = 𝑥^2 - 2

Using a table of values, we can plot some points to create a graph.

x | y
---------------
-2 | 2
-1 | 1
0 | -2
1 | -1
2 | 2

Graph of the equation 𝑦 = 𝑥^2 - 2:
- - - - - - - - - - - - -
.
.
.
As you can see, the parabola is shifted 2 units downwards compared to the graph of a standard parabola.

In summary, the constant 𝑎 determines the stretch, 𝑘 determines the vertical shift, and ℎ determines the horizontal shift of the parabola defined by the equation 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘.