How do you find the axis of symmetry, graph and find the maximum or minimum value of the function #f(x)=1/3(x+6)^2+5#?

Question

Kennedy Adams · Accepted Answer

Expand the equation and simplify it and then complete the square.

#(x^3)/3 +4x+17# This the equation you will get when you expand the bracket and simplify. #1/3(x+6)^2-5# This is the equation when you complete the square. To find the X and Y value. Take what is in the bracket and equate it to zero. So, in this case, it is X+6=0 and X=-6. Now for the Y value, the value which is in the far right is the Y value, in this case, it is -5. So the coordinates for the minimum value is (-6,-5)

Jameson Allen · Answer

undefined To find the axis of symmetry of the function $f(x) = \frac{1}{3}(x + 6)^2 + 5$, use the formula $x = -\frac{b}{2a}$, where $a = \frac{1}{3}$ and $b = 6$. The axis of symmetry is $x = -\frac{6}{2(\frac{1}{3})}$, which simplifies to $x = -6$.

The graph of the function is a parabola that opens upwards because the coefficient of $x^2$ is positive. The vertex of the parabola is at the point $(-6, 5)$, which corresponds to the axis of symmetry.

To find the maximum or minimum value of the function, observe that the coefficient of $x^2$ is positive, indicating a minimum value. The minimum value occurs at the vertex, which is $f(-6) = 5$. Therefore, the minimum value of the function is $5$.