Explain the role of a, h, and π in the equation π¦ = π(π₯ β β)2 + π. You may use words, numerical examples, and sketches to support your explanation.
The equation π¦ = π(π₯ β β)2 + π is in standard form and represents a parabola.
- The value of π is the π¦-coordinate of the vertex of the parabola.
- The value of β is the π₯-coordinate of the vertex of the parabola.
- The value of π determines the shape, direction, and steepness of the parabola.
To illustrate this, consider the following parabolas:
- If π > 0, the parabola opens upwards (like a smiley face), and its vertex is the lowest point. For example, π¦ = 2(π₯ β 3)2 + 1 has a vertex of (3,1), and the parabola looks like this:
![image.png](attachment:image.png)
- If π < 0, the parabola opens downwards (like a frown), and its vertex is the highest point. For example, π¦ = β3(π₯ + 2)2 + 4 has a vertex of (β2,4), and the parabola looks like this:
![image-2.png](attachment:image-2.png)
- If π = 0, the equation is that of a horizontal line with a π¦-intercept of π. For example, π¦ = 7 has a horizontal line with a π¦-intercept of 7.
In summary, π controls the steepness of the parabola, β shifts the parabola left or right, and π raises or lowers the parabola.
In the equation π¦ = π(π₯ β β)Β² + π, the variables π, β, and π have specific roles and influence the shape, position, and the direction of the graph of the quadratic function.
1. Role of π:
- The variable π is the coefficient of the quadratic term in the equation.
- It determines the stretch or compression of the parabola along the vertical axis.
- When π > 0, the parabola opens upwards and is stretched vertically. The larger the value of π, the narrower the parabola becomes. For example, if π = 2, the graph will be narrower than if π = 1.
- When π < 0, the parabola opens downwards and is compressed vertically. The absolute value of π determines the steepness of the parabola. For example, if π = -2, the graph will be steeper than if π = -1.
2. Role of β:
- The variable β represents the horizontal shift of the vertex of the parabola.
- The vertex of the parabola is located at the point (β, π). If β > 0, the vertex will be shifted to the right; if β < 0, the vertex will be shifted to the left. The magnitude of β determines the amount of the horizontal shift.
- For example, if β = 3, the vertex will be shifted 3 units to the right from the origin (0, 0). If β = -2, the vertex will be shifted 2 units to the left.
3. Role of π:
- The variable π represents the vertical shift of the vertex of the parabola.
- The vertex is located at the point (β, π). If π > 0, the vertex will be shifted upward; if π < 0, the vertex will be shifted downward. The magnitude of π determines the amount of the vertical shift.
- For example, if π = 5, the vertex will be shifted 5 units upward from the origin (0, 0). If π = -3, the vertex will be shifted 3 units downward.
Here is a numerical example to illustrate the role of π, β, and π using the equation π¦ = 2(π₯ β 3)Β² + 5:
- π = 2, which means the parabola opens upwards and is stretched vertically.
- β = 3, which means the vertex is shifted 3 units to the right.
- π = 5, which means the vertex is shifted 5 units upward.
Sketching the graph of this equation will help visualize the role of π, β, and π accurately.