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

Question

Emily Ammerman · Accepted Answer

#(-b)/(2a)# gives the x coordinate of the max/min point. #(-6)/(-2)=3# #x=3# is the line of symmetry. When #x=3, y=-3^2+6xx3+6# #y=-9+18+6=15# (3,15) is the vertex, as the #x^2# is negative the parabola will be #nn# shaped.

Luke Alvarez · Answer

See explanation

As this is in calculus we do the following:

#color(blue)("Determine the general shape and vertex")#

Shortcut approach:

Given: #f(x)=-x^2+6x+6 color(white)("d")->f'(x)=-2x+6=0# at the turning point.

Set #f'(x)=0=-2x+6 => x_("vertex")=6/2=3#

By substitution set #y=-(3)^2+6(3)+6 = -9+18+6 = +15#

Vertex #->(x,y)=(3,15)#

#f''(x)=-2# and as this is negative the vertex is a maximum.

So the graph is of form #nn# which is compatible with the #x^2# term being negative. This also indicates the general form of #nn#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#color(blue)("The y-intercept")#

This is at #x=0# so set #y=-x^2+6x+6 =-(0)^2+6(0)+6=6#

#y_("intercept")=6#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#color(blue)("The x-intercept")#

As the vertex is #(x,y)=(3,15)# and the general form is #nn# then the plot crosses the x-axis so #x_("intercept")# exists.

Using Completing the square:

Set #y=0=-x^2+6x+6#

#0=-1(x-3)^2+k+6#

Set #-1(-3)^2+k=0 => k= +9# giving:

#0=-1(x-3)^2+15#

#x-3=+-sqrt(15)#

#x=3+-sqrt(15)#

#x~~ +6.87298....#
#x~~-0.87298....#

William Abdalla · Answer

Just for reference: The first calculus part only using first principles.

Given: #f(x)=-x^2+6x+6# Set: #y=-x^2+6x+6" "....................Equation(1)# Increment #x# by the very small amount of #delta x# Consequently the value of #y# would have also changed by the small amount of #deltay# By substitution #Eqn(1)# becomes: #y+deltay=-(x+deltax)^2+6(x+deltax)+6# #y+deltay=-(x^2+2xdeltax+(deltax)^2)+(6x+6deltax)+6# #y+deltay=-x^2-2xdeltax + (deltax)^2+6x+6deltax+6" ". ..Eqn(1_a)# #Eqn(1_a)-Eqn(1)# #y+deltay=-x^2-2xdeltax + (deltax)^2+6x+6deltax+6 # #ul(ycolor(white)("dddd")=-x^2color(white)("ddddddddddd.d")+6xcolor(white)("ddddd")+6)larr Subtract"# #color(white)("ddd")deltay= color(white)("dd")0color(white)("d")-2xdeltax +(deltax)^2+0color(white)("d")+6deltax+0# #deltay=-2xdeltax+(deltax)^2+6deltax# Divide both sides bu #deltax# #color(white)("ddddd")(deltay)/(deltax) =-2x+color(white)("dd")deltaxcolor(white)("ddd")+6# #lim_(deltax->0)[(deltay)/(deltax)]=-2x+ubrace(lim_(deltax->0)[ deltax ]color(white)(.))color(white)("d")+6# #color(white)(dddd"d")dy/dx= color(white)("dd")-2xcolor(white)("dd.d")+0color(white)("ddddd")+6#

Emma Aldridge · Answer

undefined To find the axis of symmetry, graph, and locate the maximum or minimum value of the function $ f(x) = -x^2 + 6x + 6 $, follow these steps:

1. **Axis of Symmetry**:
   The axis of symmetry for a parabola given by the equation $ y = ax^2 + bx + c $ is given by the formula $ x = -\frac{b}{2a} $.
   For the given function $ f(x) = -x^2 + 6x + 6 $:
   $$ a = -1, \quad b = 6 $$
   Use the formula $ x = -\frac{b}{2a} $ to find the axis of symmetry.

2. **Graphing**:
   Plot the function $ f(x) = -x^2 + 6x + 6 $ on a coordinate plane. You can use points, the axis of symmetry, and any additional information to sketch the graph accurately.

3. **Maximum or Minimum Value**:
   To find the maximum or minimum value of the function, you need to determine whether the parabola opens upwards (in which case, it has a minimum value) or downwards (in which case, it has a maximum value).
   For the function $ f(x) = -x^2 + 6x + 6 $, since the coefficient of $ x^2 $ is negative (-1), the parabola opens downwards, indicating a maximum value.
   To find the maximum value, evaluate the function at the vertex (which corresponds to the axis of symmetry). Plug the value of $ x $ obtained from the axis of symmetry into the function $ f(x) $.

4. **Summary**:
   - Axis of Symmetry: Use the formula $ x = -\frac{b}{2a} $.
   - Graph: Plot the function on a coordinate plane.
   - Maximum or Minimum Value: Determine the direction of the parabola and evaluate the function at the vertex.

By following these steps, you can find the axis of symmetry, graph the function, and locate the maximum or minimum value of $ f(x) = -x^2 + 6x + 6 $.