Y=-3×2+8×+35.Identify the axis of symmetry and the vertex?

2 Answers
Mar 22, 2018

"Vertex: " (4/3, 363/9)Vertex: (43,3639)
"Axis of Symmetry: " x=4/3Axis of Symmetry: x=43

Explanation:

y=-3x^2+8x+35y=3x2+8x+35

It's important to remember that, when it comes to quadratics, there are two forms:

f(x)=ax^2+bx+cf(x)=ax2+bx+c color(blue)(" Standard Form") Standard Form
f(x)=a(x-h)^2+kf(x)=a(xh)2+k color(blue)(" Vertex Form") Vertex Form

For this problem, we can disregard the vertex form, as our equation is in the standard form.

To find the vertex of the standard form, we have to do some math:

"Vertex:"Vertex: ((-b)/(2a), f((-b)/(2a)))(b2a,f(b2a))

The y"-coordinate"y-coordinate might look a little confusing, but all it means is that you plug in the x"-coordinate"x-coordinate of the vertex back into the equation and solve. You'll see what I mean:

x"-coordinate:"x-coordinate:

((-b)/(2a))(b2a)

((-8)/(2(-3)))(82(3)) color(blue)(" Plug in "8 " for " b " and " -3 " for " a) Plug in 8 for b and 3 for a

((-8)/-6)(86) color(blue)(" "2*3=6) 23=6

((cancel(-)4)/(cancel(-)3)) color(blue)(" Simplify; negatives cancel to make positive")

x"-coordinate: " color(red)(4/3)

Now let's plug 4/3 back into every x in the original function

y=-3x^2+8x+35

y=-3(4/3)^2+8(4/3)+35 color(blue)(" Plug " 4/3 " into the " x"'s")

y= -3(16/9)+8(4/3)+35 color(blue)(" "4^2=16," " 3^2=9)

y=-48/9 +8(4/3)+35 color(blue)(" "-3*16=-48)

y=-48/9+32/3+35 color(blue)(" "8*4=32)

Let's get some common denominators to simplify this:

y=-48/9+96/9+35 color(blue)(" "32*3=96," "3*3=9)

y=-48/9+96/9+315/9 color(blue)(" "35*9=315," "1*9=9)

y=48/9+315/9 color(blue)(" "-48/9+96/9=48/9)

y=363/9 color(blue)(" "48/9+315/9=363/9)

y"-coordinate: " color(red)(363/9)

Now that we have our x and y "coordinates," we know the vertex:

"Vertex: " color(red)((4/3, 363/9)

When it comes to quadratics, the "axis of symmetry" is always the x"-coordinate" of the "vertex". Therefore:

"Axis of Symmetry: " color(red)(x=4/3)

It's important to remember that the "axis of symmetry" is always told in terms of x.

Mar 22, 2018

x=4/3," vertex "=(4/3,121/3)

Explanation:

"the equation of a parabola in "color(blue)"vertex form" is.

color(red)(bar(ul(|color(white)(2/2)color(black)(y=a(x-h)^2+k)color(white)(2/2)|)))

"where "(h,k)" are the coordinates of the vertex and a"
"is a multiplier"

"to express y in this form use "color(blue)"completing the square"

• " the coefficient of the "x^2" term must be 1"

rArry=-3(x^2-8/3x-35/3)

• " add/subtract "(1/2"coefficient of the x-term")^2" to"
x^2-8/3x

y=-3(x^2+2(-4/3)xcolor(red)(+16/9)color(red)(-16/9)-35/3)

color(white)(y)=-3(x-4/3)^2-3(-16/9-35/3)

color(white)(y)=-3(x-4/3)^2+121/3larrcolor(red)"in vertex form"

rArrcolor(magenta)"vertex "=(4/3,121/3)

"the equation of the axis of symmetry passes through the"
"vertex is vertical with equation "x=4/3