How do you find the maximum or minimum of #y-x^2+6=9x#?

Question

Hazel Anglin · Accepted Answer

The minimum is #(-4.5, -26.25)#.

The maximum or minimum of a quadratic equation is the same as the vertex. First, we have to write the equation in standard form, or #y = ax^2 + bx + c#

To do this, let's make #y# by itself. Add #x^2# and subtract #6# from both sides of the equation:
#y - x^2 + 6 quadcolor(red)(+quadx^2 quad-quad6) = 9x quadcolor(red)(+quadx^2 quad-quad6)#

#y = x^2 + 9x - 6#

To find the #x# value of the vertex, we use the formula #x = -b/(2a)#.

Let's plug in the numbers:
#x = -9/(2(1))#

#x = -4.5#

To find the #y# value of the vertex, we substitute back in the value of #x# back into the equation:
#y = x^2 + 9x - 6#

#y = (-4.5)^2 + 9(-4.5) - 6#

#y = 20.25 - 40.5 - 6#

#y = -26.25#

Therefore, using the #x# and #y# values of the vertex, we know that the vertex is at #(-4.5, -26.25)#. To verify this, let's graph it:

Jameson Allen · Answer

undefined To find the maximum or minimum of the function y = -x^2 + 6 - 9x, first, rewrite the equation in the standard form of a quadratic function, y = ax^2 + bx + c. Then, identify the coefficient of x^2 (a), coefficient of x (b), and constant term (c). Once you have these values, you can use the formula for the x-coordinate of the vertex of a parabola, which is given by x = -b / (2a). After finding the x-coordinate of the vertex, substitute it back into the original equation to find the corresponding y-coordinate. This will give you the coordinates of the vertex, which represents the maximum or minimum point of the function. Finally, determine whether the vertex corresponds to a maximum or minimum by examining the concavity of the parabola. If the coefficient of x^2 is negative (a < 0), the parabola opens downwards, indicating a maximum. If the coefficient of x^2 is positive (a > 0), the parabola opens upwards, indicating a minimum.