for equation y=2x^2+12x-10, determine the vertex, axis of symmetry, and horizontal and vertical shifts
yeah thanks for the help
To determine the vertex, axis of symmetry, and horizontal and vertical shifts for the equation y = 2x^2 + 12x - 10, we need to start by understanding the general form of a quadratic equation: y = ax^2 + bx + c.
1. Vertex: The vertex of a quadratic equation in the form y = ax^2 + bx + c is given by the coordinates (-b/2a, f(-b/2a)), where f(x) is the equation.
For our equation y = 2x^2 + 12x - 10, a = 2, b = 12, and c = -10.
The x-coordinate of the vertex is -b/2a = -12/4 = -3.
To find the y-coordinate, substitute x = -3 into the equation: y = 2(-3)^2 + 12(-3) - 10 = -28.
Therefore, the vertex is (-3, -28).
2. Axis of Symmetry: The axis of symmetry for a quadratic equation is a vertical line that passes through the vertex. It is given by the equation x = -b/2a.
For our equation y = 2x^2 + 12x - 10, the axis of symmetry is x = -12/4 = -3.
3. Horizontal Shift: A horizontal shift occurs when we add or subtract a value to x within the equation. In this case, since there is no addition or subtraction within the equation, there is no horizontal shift. The graph remains centered at x = -3.
4. Vertical Shift: A vertical shift occurs when we add or subtract a value to the entire equation. In this case, we can rewrite the equation as y = 2(x^2 + 6x) - 10, which allows us to isolate the quadratic term.
The equation inside the parentheses, x^2 + 6x, can be rewritten as (x^2 + 6x + 9) - 9, where 9 is the square of half of the coefficient of the linear term (6/2)^2.
Therefore, the equation can be rewritten as y = 2[(x + 3)^2 - 9] - 10.
The -10 outside the brackets represents a vertical shift downward by 10 units.
So, the vertical shift is -10 units.
In summary:
- The vertex of the equation y = 2x^2 + 12x - 10 is (-3, -28).
- The axis of symmetry is x = -3.
- There is no horizontal shift.
- There is a vertical shift downward by 10 units.
y= 2(x^2+6x-5)=2(x^2+6x +9) -10 -18 check that
y= 2(x+3)^2 - 28
Can you take it from here?