How do you find the axis of symmetry, and the maximum or minimum value of the function #y= -x^2-10x+7#?

Question

Julia Adams · Accepted Answer

#x=-5" and maximum value "=32# #"We require to find the vertex and determine if maximum"# #"or minimum turning point"# #"the equation of a parabola in "color(blue)"vertex form"# is. #•color(white)(x)y=a(x-h)^2+k# #"where "(h,k)" are the coordinates of the vertex and a"# #"is a multiplier"# #x=h" is the axis of symmetry"# #"to obtain this form use "color(blue)"completing the square"# #y=-(x^2+10x-7)# #color(white)(y)=-(x^2+2(5)x+25-25-7)# #color(white)(y)=-(x+5)^2+32# #color(magenta)"vertex "=(-5,32)# #"Since "a<0" then maximum turning point "nnn# #"axis of symmetry is "x=-5# #"and maximum value "=32# graph{-x^2-10x+7 [-80, 80, -40, 40]}

Eva Adamson · Answer

Axis of symmetry: #x+5=0#

Maximum value #=32# The given equation: #y=-x^2-10x+7# #y=-(x^2+10x+25)+25+7# #y=-(x+5)^2+32# #(x+5)^2=-(y-32)# The above equation is in standard form of downward parabola #(x-x_1)^2=-4a(y-y_1)# which has Axis of symmetry: #x-x_1=0# #x+5=0# Vertex #(x-x_1=0, y-y_1=0)# #(x+5=0, y-32=0)\equiv (-5, 32)# Maximum value of given quadratic function will be at the vertex #(-5, 32)# of given downward parabola: #y=-x^2-10x+7# Hence, the maximum value of given function is obtained by setting #x=-5# in the function as follows #y(-5)=-(-5)^2-10(-5)+7# #=32#

Jameson Allen · Answer

undefined To find the axis of symmetry of the function $y = -x^2 - 10x + 7$, use the formula $x = \frac{-b}{2a}$, where $a = -1$ and $b = -10$.

So, $x = \frac{-(-10)}{2(-1)} = \frac{10}{-2} = -5$.

The axis of symmetry is $x = -5$.

To find the maximum or minimum value, substitute $x = -5$ into the function:

$y = -(-5)^2 - 10(-5) + 7$

$y = -25 + 50 + 7$

$y = 32$

So, the maximum or minimum value of the function is $y = 32$, and since the coefficient of $x^2$ is negative, the function has a maximum value.