SOLUTION: can you help me understand how to find the vertex, the line of symmetry, and range with this problem for the graph of the function f(x)=-x^2+6x-8 a. find the vertex b. the

Question 77060: can you help me understand how to find the vertex, the line of symmetry, and range with this problem
for the graph of the function f(x)=-x^2+6x-8
a. find the vertex
b. the line of symmetry
c. find the range
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
a)
First factor out the negative 1

Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form

Start with the given equation

Subtract from both sides

Factor out the leading coefficient

Take half of the x coefficient to get (ie ).

Now square to get (ie )

Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of does not change the equation

Now factor to get

Distribute

Multiply

Now add to both sides to isolate y

Combine like terms

Now the quadratic is in vertex form where , , and . Remember (h,k) is the vertex and "a" is the stretch/compression factor.

Check:

Notice if we graph the original equation we get:

Graph of . Notice how the vertex is (,).

Notice if we graph the final equation we get:

Graph of . Notice how the vertex is also (,).

So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.

So if we multiply everything by -1 we get

So the vertex is (3,1)

b)
The line of symmetry will go through the vertex so the equation is
x=3

c)
The range is any value y can be, so by our graph we can see the range is
(

, 1]
Here's our graph
graph of
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