Identify the vertex the axis of symmetry,the maximum, or minimum value and the range of each parabola
Y=x^2-4x+1
are you using "completing the square", if so ...
y = x^2 - 4x + 4 - 4+1
= (x-2)^2 - 3
I assume you can read the information you need from this form
To identify the vertex, axis of symmetry, maximum or minimum value, and range of a parabola in the form of y = ax^2 + bx + c, we can use the vertex formula and the properties of a parabola.
The given equation is y = x^2 - 4x + 1. Comparing this with the form y = ax^2 + bx + c, we can see that a = 1, b = -4, and c = 1.
1. Vertex:
The vertex formula for a parabola is given by x = -b / (2a). Substituting the values of a and b, we get x = -(-4) / (2 * 1) = 4 / 2 = 2. To find the corresponding y-coordinate, substitute this x-value back into the equation: y = (2)^2 - 4(2) + 1 = 4 - 8 + 1 = -3. Therefore, the vertex is (2, -3).
2. Axis of Symmetry:
The axis of symmetry is a vertical line passing through the vertex. In this case, the axis of symmetry is x = 2.
3. Maximum or Minimum Value:
Since the coefficient of x^2 is positive (a = 1), the parabola opens upward and has a minimum value. The y-coordinate of the vertex represents this minimum value. In this case, the minimum value is -3.
4. Range:
The range of the parabola is the set of all possible y-values that the parabola can take. Since the parabola opens upward and has a minimum value of -3, the range is all real numbers greater than or equal to -3. In interval notation, the range is [-3, ∞).
To summarize:
- Vertex: (2, -3)
- Axis of symmetry: x = 2
- Minimum value: -3
- Range: [-3, ∞)