SOLUTION: For the parabola whose equation is​ given, determine whether it opens upward or​ downward, find the​ vertex, and find the​ x- and​ y-intercepts. Then

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: For the parabola whose equation is​ given, determine whether it opens upward or​ downward, find the​ vertex, and find the​ x- and​ y-intercepts. Then      Log On


   

Question 1048919: For the parabola whose equation is​ given, determine whether it opens upward or​ downward, find the​ vertex, and find the​ x- and​ y-intercepts. Then graph the parabola.
​f(x)=x squared minus 2 x minus 48
Found 3 solutions by stanbon, MathLover1, advanced_Learner:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
For the parabola whose equation is​ given, determine whether it opens upward or​ downward, find the​ vertex, and find the​ x- and​ y-intercepts. Then graph the parabola.
​f(x)=x squared minus 2 x minus 48
---
f(x) = x^2 - 2x - 48
a = 1 ; b = -2 ; c = -48
-----
opens upward because a is positive
-------
Vertex::
x = -b/(2a) = 2/2 = 1 ; f(1) = 1^2-2*1-48 = -49
Vertex is (1,-49)
-------------------
x-intercept::
Let y = 0 ; solve for "x"::
x^2-2x-48 = 0
(x-8)(x+6) = 0
x-intercepts:: x = 8 ; x = -6
---------------------------
y-intercept::
Let x = 0 ; solve for "y"::
y = 0^2 - 2*0 - 48
y = -48
----------------
Graph::
graph%28400%2C400%2C-20%2C20%2C-60%2C60%2Cx%5E2+-+2x+-+48%29
------------
Cheers,
Stan H.
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Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29=x%5E2+-x+-+48
first write it in vertex form f%28x%29=a%28x-h%29%5E2+%2Bk where h and k are x and y coordinates of the vertex
f%28x%29=%28x%5E2+-x%29+-+48...complete square
f%28x%29=%28x%5E2+-x%2Bb%5E2%29-b%5E2+-+48.....since coefficient a=1 and 2ab=-1, find b
2%2A1b=-1->2b=-1->b=-1%2F2
so, we have
f%28x%29=%28x%5E2+-x-%28-1%2F2%29%5E2%29-%28-1%2F2%29%5E2+-+48
f%28x%29=%28x-1%2F2%29%5E2-%281%2F4%29+-+48
f%28x%29=%28x-1%2F2%29%5E2-1%2F4+-+192%2F4
f%28x%29=%28x-1%2F2%29%5E2+-+193%2F4

so, h=1%2F2 and k=-193%2F4
or h=0.5 and k=-48.25

set f%28x%29=0 to find x-intercepts
0=%28x-1%2F2%29%5E2+-+193%2F4
193%2F4=%28x-1%2F2%29%5E2+
sqrt%28193%2F4%29=x-1%2F2+
1%2F2%2Bsqrt%28193%29%2F2=x+
x-intercepts are:
x=1%2F2%2Bsqrt%28193%29%2F2 or x=1%2F2-sqrt%28193%29%2F2+
x=7.5 or x=-6.5%29


set x=0 to find y-intercepts
f%28x%29=%280-1%2F2%29%5E2+-+193%2F4
f%28x%29=%28-1%2F2%29%5E2+-+193%2F4
f%28x%29=1%2F4+-+193%2F4
f%28x%29=+-+192%2F4
f%28x%29=+-+48 which is same if you use f%28x%29=x%5E2+-x+-+48->f%28x%29=0%5E2+-0+-+48=-48
x=7.5 or x=-6.5%29

Answer by advanced_Learner(501) About Me  (Show Source):
You can put this solution on YOUR website!
given function
y=x%5E2-2x-48
find
a.opens upward or downward?
b.vertex
c.x intercepts?
d.y-intercepts.
e.graph
Solution
a

y=x%5E2-2x-48
a=1,b=-2,c=-48
therefore,it opens upward since a=1 is positive.

b.
y=x%5E2-2x-48
vertex x part =-b%2F2a
vertex=2%2F2
vertex=1
vertex y part =1-2-48
vertex y part=-49
another solution by completing the square.
48=x%5E2-2x
48%2B1=x%5E2-2x%2B1
y=%28x-1%29%5E2-49 so now the function is written as vertex form
y=a%28x-h%29%5E2+k
so clearly
h=1
k=-49

c.x intercepts
y=x%5E2-2x-48
set y=0 and solve for x
0=x%5E2-2x-48
0=%28x%2B6%29%28x-8%29
x=8 or x=-6



Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-2x%2B-48+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-2%29%5E2-4%2A1%2A-48=196.

Discriminant d=196 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--2%2B-sqrt%28+196+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-2%29%2Bsqrt%28+196+%29%29%2F2%5C1+=+8
x%5B2%5D+=+%28-%28-2%29-sqrt%28+196+%29%29%2F2%5C1+=+-6

Quadratic expression 1x%5E2%2B-2x%2B-48 can be factored:
1x%5E2%2B-2x%2B-48+=+1%28x-8%29%2A%28x--6%29
Again, the answer is: 8, -6. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-2%2Ax%2B-48+%29





d.y intercepts
set x=0 and solve for y
y=x%5E2-2x-48
y=0-0-48
y=-48



e.graph
y=x%5E2-2x-48
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-2x%2B-48+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-2%29%5E2-4%2A1%2A-48=196.

Discriminant d=196 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--2%2B-sqrt%28+196+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-2%29%2Bsqrt%28+196+%29%29%2F2%5C1+=+8
x%5B2%5D+=+%28-%28-2%29-sqrt%28+196+%29%29%2F2%5C1+=+-6

Quadratic expression 1x%5E2%2B-2x%2B-48 can be factored:
1x%5E2%2B-2x%2B-48+=+1%28x-8%29%2A%28x--6%29
Again, the answer is: 8, -6. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-2%2Ax%2B-48+%29