Algebra 2 help!!!? plz!?

Question

Can someone show me the steps on how to do this promblem!? please!!!

Find the coordinates of the vertex of the graph of the function. Tell whether the vertex is a maximum or a minimum.

y = -4x^2 + 4x + 2

Answer 1

dy/dx= -8x+4 which =0 at a turning point

-8x+4=0
x=1/2

sub x=1/2 into y

y= -1+2+2=3
so (1/2, 3) is a vertex.

to characterise the turning point, find d2y/dx2

d2y/dx2= -8

it is a negative number means the graph is an upside down parabola. therefore the vertex is a maximum

Answer 2

This is simple if you go about it the right way.

For a start, there is a simple equation for the 'axis of symmetry' for a quadratic expressed in the form (ax^2 + bx + c). Which yours is.

The axis of symmetry is: x = -b / 2a

Note that this is a vertical line - a quadratic is always symmetrical about a vertical line, right?

For your equation, x = -b / 2a = 4 / (2 * 4) = 1/2.

To find the vertex, just substitute the value of x.

y = -4x^2 + 4x + 2 = -4 * .25 + 4 * .5 + 2 = -1 + 2 + 2 = 3

Therefore, the vertex is (0.5, 3)

You can tell whether the vertex is a minimum or a maximum by the sign of 'a'. The simple quadratic y = x^2 is 'concave upwards' and it has a minumum at (0,0). As x increases (in the positive or negative direction), you can see that y gets larger. Therefore the vertex is a minimum.

If the value of 'a' is negative (as it in your case: -4) you can see that as 'x' gets large, y gets large in the *negative* direction. Therefore the vertex is a maximum.

You can test this simply by putting 'x=0' into the equation. You get 'y = 2', which is a lower y value than the vertex. The vertex can only be a maximum or a minimum, therefore in this case it must be a maximum.

Hope this is clear.

Answer 3

Quadratic functions are always parabolas. The negative sign in front of the quadratic term (-4x^2) says that it opens downward.

How do you find the vertex?

Here's my own special approach:
First, ignore the 2. (Fancy way to say this is translate--or slide--function down by 2.) We have
y=-4x^2+4x.
Factor out 4x:
y=4x(-x+1)

This thing has roots at x=0 and x=1. That means the axis of symmetry is at the average, x=.5. We plug that into the function to find y=-4(.25)+4(.5)+2 = -1+2+2=3.

The vertex is at (.5, 3) and 3 is the maximum value; all other points are lower.

A shortcut for finding the vertex is "x=-b/(2a)". In this case a means the quadratic coefficient -4; b means the linear coefficient 4. The 2 is commonly called c, but the shortcut ignores c (just like we did earlier). So, the vertex happens at x=-4/-8=.5.



One last approach:
y=-4x^2+4x+2
=3-1+4x-4x^2
=3-(1-4x+4x^2)
y=3-(1-2x)^2

Since we have 3 minus (something)^2, the biggest value we can get out of this is 3, which happens when the (something) is 0. That is, when 1-2x=0, or 1=2x, or x=.5. Thus, the vertex at (.5,3) is the max.

Answer 4

y=-4x^2+4x+2
y-2 =-4x^2+4x----------Move the 2 over to the left
y-2 = -4(x^2-x) ---------Factor out the -4
y-2 -1 = -4(x^2 -x + 1/4) Complete the square half of -1 squared is 1/4. On the left you have to add -1 because the 1/4 on the right is added under the parenthesis and has the effect of being multiplied by the -4 so to keep the equation balanced you need to add -1 to the left.
y-3= -4(x-1/2)^2 Factor the right side
y = -4(x-1/2)^2 +3 Set = to y

the vertex is (1/2,3) Since the 4 is negative the parabola opens downward and that makes the vertex a maximum.

Answer 5

(y - k) = a(x - h)^2
y = -4(x^2 - x - 1/2)
y = -4(x^2 - x + 1/4 - 1/4 - 1/2)
y = -4(x - 1/2)^2 - 3/4)
y = -4(x - 1/2)^2 + 3
(y - 3) = -4(x - 1/2)^2

The vertex is at (1/2, 3), and the parabola opens down, so the vertex is a maximum.

Answer 6

Since the coefficient of x^2 is -ve, this means the parabola opens downwards and the value will be maximum.

You dont need to derive the result for coordinates of vertex and waste time,it's coordinates are given by:(for y=ax^2 + bx +c)

-b/(2a) , D/(4a)

[D=discriminant of f(x)]

here y maximum is D/(4a) at x=-b/(2a)

Answer 7

well your vertex is at x=.5 and y=3

because of the negative sign on 4x^2 you know the parabola is curved down so the vertex would be a maximum.

As far as the math, I couldn't tell you how to get it into certain forms.

Use www.wikipedia.com to look up formulas.

Answer 8

a million. on condition that -a million does no longer contain the variable to clean up for, flow it to the wonderful-hand area of the equation via including a million to the two factors. 9x^(2)=a million 2. Divide each and each term interior the equation via 9. (9x^(2))/(9)=(a million)/(9) 3. Simplify the left-hand area of the equation via canceling the user-friendly factors. x^(2)=(a million)/(9) 4. Take the sq. root of the two factors of the equation to do away with the exponent on the left-hand area. x=~((a million)/(9)) 5. split the fraction interior the unconventional right into a separate radical expression interior the numerator and the denominator. a fragment of roots is similar to a root of the fraction. x=(a million)/(~(9)) 6. Pull all ideal sq. roots out from under the unconventional. for this reason, get rid of the three because of the fact it rather is a ideal sq.. x=(a million)/(3) 7. First, exchange interior the + ingredient of the to discover the 1st answer. x=(a million)/(3) 8. next, exchange interior the - ingredient of the to discover the 2nd answer. x=-(a million)/(3) 9. the finished answer is the effect of the two the + and - parts of the answer. answer: x=(a million)/(3),-(a million)/(3) wish this helps u :)