without graphing, find the vertex, the azis of symemtry, and the maximum value or the minium value.
sorry i forgot to put the problem. here it is:
f(x)=4(x+5)^2-6
y-k = a(x-h)^2
has vertex at (h,k).
Naturally, the max/min is at the vertex and the vertex is on the axis of symmetry.
To find the vertex, the axis of symmetry, and the maximum or minimum value of a quadratic equation without graphing, you can use the process of completing the square.
1. Start with the quadratic equation in the standard form: f(x) = ax^2 + bx + c.
2. Determine the coefficient values of a, b, and c from the given quadratic equation.
3. Calculate the x-coordinate of the vertex using the formula: x = -b / (2a). This will give you the x-coordinate of the vertex and the axis of symmetry.
4. Substitute the x-coordinate of the vertex you calculated in Step 3 into the original quadratic equation to find the corresponding y-coordinate. This will give you the y-coordinate of the vertex, which represents the minimum or maximum value of the quadratic function.
5. The axis of symmetry is the vertical line that passes through the x-coordinate of the vertex. Its equation can be written as x = (value of x-coordinate from Step 3).
6. Depending on the coefficient value of a, the parabola will either open upwards (if a > 0) or downwards (if a < 0). If a > 0, the y-coordinate of the vertex is the minimum value of the quadratic function. If a < 0, the y-coordinate of the vertex is the maximum value of the quadratic function.
By following these steps, you can find the vertex, axis of symmetry, and the maximum or minimum value of a quadratic equation without graphing it.