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y = -2x^2 + 13x - 6
Does the graph of the function open up or down?

2006-10-15 06:09:08 · 3 answers · asked by racheldawnsmith311 1 in Science & Mathematics Mathematics

3 answers

y-intercept: set x=0, and you get y=-6.
x-intercept: set y=0, and you get the equation
2x^2 - 13x + 6 = 0, which factors into
(2x-1)(x-6) = 0.
Therefore x=1/2 and x=6 are the x-intercepts.

domain: all real numbers (a polynomial is defined everywhere)
range: complete the square and get
y = -2(x^2 - 13x/2 + 3)
= -2[(x - 13/4)^2 + 121/16]
= -2(x-13/4)^2 + 121/8.

-2(x-13/4)^2 <= 0, so y = -2(x-13/4)^2 + 121/8 <= 121/8.

Because the leading coefficient is negative, it opens down.

2006-10-15 06:46:40 · answer #1 · answered by James L 5 · 0 0

the x intercept occurs when y=0
0= -2x^2 +13x -6
0= 2x^2 -13x +6
factor the trinomial
(2x -1)(x-6)=0
set each factor equal to 0 and solve
2x-1=0 x-6=0
2x=1 x=6
x=1/2
x-intercepts are at (1/2,0) and (6,0)

the y-intercept happens when x=0
y= -2(0)^2 +13(0) -6
the first two terms go to 0 so the result is:
y= -6

the y intercept is at (0,-6)

the vertex occurs at half the distance between the x- intercepts:
add the two x- intercept values together and divide by two:
(1/2+6)/2= (13/2)/2= 13/4. This is the x-value for the vertex.
{if you know calculus simply take the first derivative and set it equal to 0 to find the x-value)
now substitute this value for x into the original funcion and solve for y to find the y-value for the vertex
y= -2(13/4)^2 +13(13/4) -6

y= 121/8 or 15 1/8

vertex is at (13/4, 121/8)
the domain of the function is from negative infinity to positive infinity. The range is from negative infinity to 121/8 since it cannot go above the vertex. It includes the value 121/8
Your last question is the easiest to answer. Since the coefficient on the squared term is negative, it opens down, like an upside down coffee cup. You can know this without even graphing it.

2006-10-15 07:10:21 · answer #2 · answered by Uncle Bill 2 · 0 0

y(x)=Ax^2+Bx+C

vertex @ (-B/2A, y(-B/2A)) so we have (13/4,121/8)

x intercepts found by setting y(x)=0 & solve for x
x={1/2,6} use factoring set each factor =0

y intercept @(0,y(0)) so we have (0,-6)

domain is all reals range is (-infinity, 121/8) since vertex is max y value

all quadratics with negative value for A open down

2006-10-15 06:55:03 · answer #3 · answered by ivblackward 5 · 0 0

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