How do you find the axis of symmetry, and the maximum or minimum value of the function #y=x^2+3x-5#?

Answer 1

The axis of symmetry is #x=-3/2# or #-1.5#.

The vertex is #(-3/2,-29/4)# or #(-1.5,-7.25)#.

Given:

#y=x^2+3x-5# is a quadratic equation in standard form:
#y=ax^2+bx+c#,

where:

#a=1#, #b=3#, #c=-5#
Axis of symmetry: vertical line that divides the parabola into two equal halves. It is also the #x#-coordinate of the vertex.

The formula to find the axis of symmetry:

#x=(-b)/(2a)#

Plug in the known values.

#x=(-3)/(2*1)#
#x=-3/2# or #-1.5#
The axis of symmetry is #x=-3/2# or #-1.5#.
Vertex: the minimum or maximum point on the parabola. The axis of symmetry is the #x#-coordinate. To find the #y#-coordinate, substitute #-3/2# into the equation and solve for #y#.
#y=(-3/2)^2+3(-3/2)-5#
#y=9/4-9/2-5#
Multiply #9/2# by #2/2#, and #5# by #4/4# so each term has #4# as its denominator.
#y=9/4-9/2xxcolor(red)2/color(red)2-5xxcolor(green)4/color(green)4#
#y=9/4-18/4-20/4#

Simplify.

#y=((9-18-20))/4#
#y=-29/4#
The vertex is #(-3/2,-29/4)# or #(-1.5,-7.25)#.

graph{y=x^2+3x-5 [-16.02, 16.01, -8.01, 8.01]}

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Answer 2

The axis of symmetry of the function ( y = x^2 + 3x - 5 ) is given by the formula ( x = -\frac{b}{2a} ), where ( a ) is the coefficient of the quadratic term and ( b ) is the coefficient of the linear term. In this case, ( a = 1 ) and ( b = 3 ), so the axis of symmetry is ( x = -\frac{3}{2} ).

To find the maximum or minimum value of the function, we can evaluate the function at the axis of symmetry. Substitute ( x = -\frac{3}{2} ) into the function to get ( y = (-\frac{3}{2})^2 + 3(-\frac{3}{2}) - 5 = -\frac{17}{4} ).

Therefore, the axis of symmetry is ( x = -\frac{3}{2} ), and the function has a maximum value of ( y = -\frac{17}{4} ) at that point.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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