Given the function f(x)=2x^2 - 4x
What is the orientation, vertex, axis of symmetry equation, and x & y intercepts.
f(x)= 2(x-1)^2-2 (complete the square to get vertex form f(x)=a(x-h)+k)
Vertex is (h,k) so (1,-2)
Axis of sym = 1
x intercepts = (0,0) and (2,0)
y intercept = 0
parabola opens up
y is the vertex not (-1,-2)?
To determine the orientation, vertex, axis of symmetry equation, and x and y intercepts of the function f(x) = 2x^2 - 4x, we can analyze its properties step by step:
1. Orientation: The coefficient of x^2 is positive (2 > 0), so the function opens upward. This means the graph of the function will be a U-shaped curve with a minimum turning point.
2. Vertex: The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a and b are the coefficients of x^2 and x, respectively. For f(x) = 2x^2 - 4x, a = 2 and b = -4. Substituting these values into the formula, we get x = -(-4) / (2 * 2) = 4 / 4 = 1. Therefore, the x-coordinate of the vertex is 1.
To find the y-coordinate of the vertex, substitute the x-coordinate (1) back into the original equation: f(1) = 2(1)^2 - 4(1) = 2 - 4 = -2. Hence, the y-coordinate of the vertex is -2.
Therefore, the vertex of the function is (1, -2).
3. Axis of Symmetry Equation: The axis of symmetry is a vertical line that passes through the vertex. Since the x-coordinate of the vertex is 1, the equation of the axis of symmetry is x = 1.
4. x-intercepts: To find the x-intercepts, set f(x) equal to zero and solve for x:
2x^2 - 4x = 0
2x(x - 2) = 0
This equation is satisfied if either 2x = 0 or (x - 2) = 0. Solving these equations, we find:
2x = 0 -> x = 0
x - 2 = 0 -> x = 2
Therefore, the x-intercepts of the function are (0, 0) and (2, 0).
5. y-intercept: To find the y-intercept, substitute x = 0 into the original equation:
f(0) = 2(0)^2 - 4(0) = 0 - 0 = 0
Thus, the y-intercept of the function is (0, 0).
To summarize:
- The orientation is upward (opens upward).
- The vertex is at (1, -2).
- The equation of the axis of symmetry is x = 1.
- The x-intercepts are (0, 0) and (2, 0).
- The y-intercept is (0, 0).
To determine the orientation, vertex, axis of symmetry, and x & y intercepts of the given function f(x) = 2x^2 - 4x, we can follow these steps:
1. Orientation:
The orientation of a quadratic function is determined by the coefficient of the x^2 term. If the coefficient is positive (greater than 0), the parabola opens upwards, indicating a concave up orientation. If the coefficient is negative (less than 0), the parabola opens downwards, indicating a concave down orientation. In this case, the coefficient of x^2 is 2, which is positive, so the parabola opens upwards.
2. Vertex:
The vertex of a quadratic function represents the highest or lowest point on the graph of the parabola. The x-coordinate of the vertex can be found using the formula: x = -b / (2a), where a and b are the coefficients of the x^2 and x terms, respectively. For our function f(x) = 2x^2 - 4x, a = 2 and b = -4. Thus, using the formula, we find x = -(-4) / (2 * 2) = 4 / 4 = 1. This means that the x-coordinate of the vertex is 1.
To find the y-coordinate of the vertex, substitute the x-coordinate back into the function: f(1) = 2(1)^2 - 4(1) = 2 - 4 = -2. Therefore, the vertex of the function is (1, -2).
3. Axis of Symmetry:
The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two congruent halves. Its equation can be determined from the x-coordinate of the vertex. In this case, the axis of symmetry is x = 1.
4. x-intercepts:
To find the x-intercepts, set f(x) equal to zero and solve for x. In this case, we have 2x^2 - 4x = 0. Factoring out x, we get x(2x - 4) = 0. This gives us two solutions: x = 0 and 2. Therefore, the x-intercepts are (0, 0) and (2, 0).
5. y-intercept:
To find the y-intercept, substitute x = 0 into the function f(x) = 2x^2 - 4x. This gives us f(0) = 2(0)^2 - 4(0) = 0. Therefore, the y-intercept is (0, 0).
In summary:
- Orientation: The parabola opens upwards (concave up).
- Vertex: The vertex is located at (1, -2).
- Axis of Symmetry: The equation of the axis of symmetry is x = 1.
- x-intercepts: The x-intercepts are (0, 0) and (2, 0).
- y-intercept: The y-intercept is (0, 0).