SOLUTION: please help me find the roots, lines of symmetry, and vertex for y=-x2+5x-4

Algebra ->  Algebra  -> Quadratic Equations and Parabolas -> SOLUTION: please help me find the roots, lines of symmetry, and vertex for y=-x2+5x-4      Log On

Ad: You enter your algebra equation or inequality - Algebrator solves it step-by-step while providing clear explanations. Free on-line demo .
Ad: Algebrator™ solves your algebra problems and provides step-by-step explanations!
Ad: Algebra Solved!™: algebra software solves algebra homework problems with step-by-step help!

   

Question 414468: please help me find the roots, lines of symmetry, and vertex for y=-x2+5x-4
Found 3 solutions by stanbon, rfer, MathLover1:
Answer by stanbon(57347) About Me  (Show Source):
You can put this solution on YOUR website!
please help me find the roots, lines of symmetry, and vertex for
y=-x^2+5x-4
----
-x^2+5x+? = y+4+?
----
Complete the square on the x-terms:
-(x^2-5+(5/2)^2) = y+4-(5/2)^2
---
Factor:
-(x-(5/2))^2 = y + 16/4 - (25/4)
----
-(x-(5/2))^2 = y -9/4
-------
Symmetry line : x = 5/2
Vertex: (5/2 , (9/4))
----
Roots of y = -x^2+5x-4
Solve -x^2+5x-4 = 0
x^2-5x+4 = 0
(x-4)(x-1) = 0
x = 1 or x = 4
---
Cheers,
Stan H.

Answer by rfer(12660) About Me  (Show Source):
You can put this solution on YOUR website!
zero's =(1,0), (4,0)
vertex=(2.5,2.5)
axis=2.5

Answer by MathLover1(6634) About Me  (Show Source):
You can put this solution on YOUR website!
y=-x%5E2%2B5x-4....factor
y=-x%5E2%2Bx%2B4x-4
y=%28-x%5E2%2Bx%29%2B%284x-4%29
y=-x%28x-1%29%2B4%28x-1%29
y=%28x-1%29%284-x%29

This is a quadratic equation (degree of 2), so
First, find x-intercepts,
Let+y+=+0 and solve for x:
%28x-1%29%284-x%29=0
x-1=0........->....x+=+1 and
4-x=0........->....x+=+4
So, the x-intercepts (roots) are (4,0) and (1,0)

Now, find the y-intercepts, let x+=+0 and solve for y,
%280-1%29%284-0%29=-4, so the y-intercept is (0,-4)

The axis of symmetry is just the average of the two x-intercepts,
%284+%2B+1%29%2F2=+2.5
So, the axis of symmetry is x+=+2.5
Since the sign of x%5E2 is negative then this function will have a maximum.
Now, to find it's maximum (vertex).
Formula for maximum value is x+=+-b%2F2a
a+=+-1, b+=+5, c+=+-4 ( Coefficients of y+=+-x%5E2+%2B+5x-+4+)
x+=+-5+%2F+%282%2A-1%29+=+5%2F2
this is exactly 2.5, which is not surprising because the maximum+occurs on the axis of symmetry.
Now, find the y value corresponding to the x value,
Substitute, x+=+2.5 into the equation and solve for y,
y=%282.5-1%29%284-2.5%29
y+=+%281.5%29%281.5%29
y+=+2.25

Summarize:
You should have more than enough now for a graph,
x-intercepts: (4,0) and (1,0)
y-intercept: (0,-4)
maximum value (vertex) : (2.5,2.25)
axis of symmetry: x+=+2.5

+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+5%2C+-x%5E2%2B5x-4%29+