SOLUTION: consider the functions f(x)=-x^2+3x+10 and g(x)=2x^2+2x+11/4. what is the exact distance between the vertices of the graphs of these two functions? cannot use graphing to answer.

Question 246034: consider the functions f(x)=-x^2+3x+10 and g(x)=2x^2+2x+11/4. what is the exact distance between the vertices of the graphs of these two functions? cannot use graphing to answer.
hopefully this will be my last question of the year.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Part 1) Find the vertices of and

part a) Let's find the vertex of

In order to find the vertex, we first need to find the x-coordinate of the vertex.

To find the x-coordinate of the vertex, use this formula: .

Start with the given formula.

From , we can see that , , and .

Plug in and .

Multiply 2 and to get .

Reduce.

So the x-coordinate of the vertex is . Note: this means that the axis of symmetry is also .

Now that we know the x-coordinate of the vertex, we can use it to find the y-coordinate of the vertex.

Start with the given equation.

Plug in .

Square to get .

Multiply and to get .

Multiply and to get .

Combine like terms.

So the y-coordinate of the vertex is .

So the vertex is

.

------------------------

b) Now let's find the vertex of

In order to find the vertex, we first need to find the x-coordinate of the vertex.

To find the x-coordinate of the vertex, use this formula: .

Start with the given formula.

From , we can see that , , and .

Plug in and .

Multiply 2 and to get .

Reduce.

So the x-coordinate of the vertex is . Note: this means that the axis of symmetry is also .

Now that we know the x-coordinate of the vertex, we can use it to find the y-coordinate of the vertex.

Start with the given equation.

Plug in .

Square to get .

Multiply and to get .

Multiply and to get .

Combine like terms.

So the y-coordinate of the vertex is .

So the vertex is

.

--------------------------------------------------

So to recap, the vertices of and are

and

respectively.

===============================================================

Part 2) Now use the distance formula to find the distance between the two vertices (which are essentially points)

Note:

is the first point

. So this means that and .
Also,

is the second point

. So this means that and .

Start with the distance formula.

Plug in , , , and .

Subtract from to get .

Subtract from to get .

Square to get .

Square to get .

Add to to get .

Simplify the square root.

So our answer is

So the exact distance between the two vertices is units.
RELATED QUESTIONS
Consider the two functions: f(x)=x^2-4 and g(x)=2x^3-3x^2-5x+6 A. A function is defined... (answered by greenestamps)

Consider the functions {{{f(x) = 3x^2 + 1}}} and {{{g(x) = x + 3}}}. What is the value of (answered by kingme18)

Grap the function:... (answered by Fombitz)

given functions f and g, determine the domain of f+g f(x)=3x^2 -9 g(x)=2x^3... (answered by nyc_function)

Let f(x) = x^4-3x^2 + 2 and g(x) = 2x^4 - 6x^2 + 2x -1. What is the degree of f(x)*... (answered by greenestamps)

Consider the functions f(x) = 4x + 2 and g(x) = 5x − 2. What is the value of ( f... (answered by Gogonati)

What is the value of g(x) - f(x) if: f(x) = 2x + 4 g(x) = 3x - 2 x =... (answered by Theo)

What is the value of g(x) - f(x) if f(x) = 2x + 4, g(x) = 3x - 2 and x = 12? 3x - 2 -... (answered by Alan3354)

b) Given the functions f(x) is 3x^2-5 and g(x) is 4+2x, find the following (i) (fog)... (answered by Boreal)