what characteristics will two parabolas in the family f(x)= a(x-3)(x+4) share?
same zeroes
open up or down depending on the sign of a
vertex lies on the line x = 1/2 (midway between the roots)
Two parabolas in the family represented by the equation f(x) = a(x - 3)(x + 4) will share the following characteristics:
1. Vertex: The vertex of both parabolas will be at the point (3, 0) since the x-coordinate of the vertex is the average of the zeros or x-intercepts, which are 3 and -4.
2. Axis of Symmetry: The axis of symmetry will be the vertical line passing through the vertex, which is x = 3 for both parabolas.
3. Shape: Both parabolas will have the same basic shape, which is determined by the quadratic term (x^2).
4. Direction: The parabolas will open upwards or downwards depending on the value of the coefficient 'a'. If a > 0, the parabolas will open upwards, and if a < 0, they will open downwards.
5. Stretch or Compression: The value of 'a' determines the stretch or compression of the parabolas. If |a| > 1, the parabolas will be stretched vertically, and if 0 < |a| < 1, they will be compressed vertically.
6. Zeros or x-intercepts: Both parabolas will have x-intercepts at x = 3 and x = -4 since those values make the expression (x - 3)(x + 4) equal to zero.
7. y-intercept: The y-intercept will be at the point (0, -12a) since substituting x = 0 into the equation gives f(0) = a(0 -3)(0 + 4) = -12a.
To determine the characteristics that two parabolas in the family f(x) = a(x-3)(x+4) share, we need to identify the common attributes that all parabolas in this family possess.
1. Vertex: The vertex is the point at which the parabola reaches its minimum or maximum value. For the parabolas in the given family, the vertex is at (3, 0). Regardless of the value of 'a', all parabolas in this family will have the same vertex.
2. Axis of Symmetry: The axis of symmetry is the vertical line that passes through the vertex of the parabola, dividing it into two equal halves. In this case, the axis of symmetry is x = 3. So, all parabolas in this family will have the same axis of symmetry.
3. Shape: The shape of a parabola depends on the sign of the coefficient 'a'. If 'a' is positive, the parabola opens upwards, resembling a "U" shape. If 'a' is negative, the parabola opens downwards, resembling an "n" shape. In this family of parabolas, regardless of the value of 'a', the shape of the parabola remains the same.
4. x-intercepts: To find the x-intercepts of a parabola, we set f(x) equal to zero and solve for 'x'. In this case, we have f(x) = a(x-3)(x+4) = 0. The x-intercepts occur at x = 3 and x = -4 for all the parabolas in this family.
In summary, the two parabolas in the family f(x) = a(x-3)(x+4) share the following characteristics:
- Same vertex at (3, 0).
- Same axis of symmetry, x = 3.
- Same shape: either a "U" shape if 'a' is positive or an "n" shape if 'a' is negative.
- Same x-intercepts at x = 3 and x = -4