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.

Algebra ->  Functions -> 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.       Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!
Part 1) Find the vertices of f%28x%29=-x%5E2%2B3x%2B10 and g%28x%29=2x%5E2%2B2x%2B11%2F4

part a) Let's find the vertex of f%28x%29=-x%5E2%2B3x%2B10



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: x=%28-b%29%2F%282a%29.


x=%28-b%29%2F%282a%29 Start with the given formula.


From y=-x%5E2%2B3x%2B10, we can see that a=-1, b=3, and c=10.


x=%28-%283%29%29%2F%282%28-1%29%29 Plug in a=-1 and b=3.


x=%28-3%29%2F%28-2%29 Multiply 2 and -1 to get -2.


x=3%2F2 Reduce.


So the x-coordinate of the vertex is x=3%2F2. Note: this means that the axis of symmetry is also x=3%2F2.


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


y=-x%5E2%2B3x%2B10 Start with the given equation.


y=-%283%2F2%29%5E2%2B3%283%2F2%29%2B10 Plug in x=3%2F2.


y=-1%289%2F4%29%2B3%283%2F2%29%2B10 Square 3%2F2 to get 9%2F4.


y=-9%2F4%2B3%283%2F2%29%2B10 Multiply -1 and 9%2F4 to get -9%2F4.


y=-9%2F4%2B9%2F2%2B10 Multiply 3 and 3%2F2 to get 9%2F2.


y=49%2F4 Combine like terms.


So the y-coordinate of the vertex is y=49%2F4.


So the vertex is .


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b) Now let's find the vertex of g%28x%29=2x%5E2%2B2x%2B11%2F4




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: x=%28-b%29%2F%282a%29.


x=%28-b%29%2F%282a%29 Start with the given formula.


From y=2x%5E2%2B2x%2B11%2F4, we can see that a=2, b=2, and c=11%2F4.


x=%28-%282%29%29%2F%282%282%29%29 Plug in a=2 and b=2.


x=%28-2%29%2F%284%29 Multiply 2 and 2 to get 4.


x=-1%2F2 Reduce.


So the x-coordinate of the vertex is x=-1%2F2. Note: this means that the axis of symmetry is also x=-1%2F2.


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


y=2x%5E2%2B2x%2B11%2F4 Start with the given equation.


y=2%28-1%2F2%29%5E2%2B2%28-1%2F2%29%2B11%2F4 Plug in x=-1%2F2.


y=2%281%2F4%29%2B2%28-1%2F2%29%2B11%2F4 Square -1%2F2 to get 1%2F4.


y=1%2F2%2B2%28-1%2F2%29%2B11%2F4 Multiply 2 and 1%2F4 to get 1%2F2.


y=1%2F2-1%2B11%2F4 Multiply 2 and -1%2F2 to get -1.


y=9%2F4 Combine like terms.


So the y-coordinate of the vertex is y=9%2F4.


So the vertex is .


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So to recap, the vertices of f%28x%29=-x%5E2%2B3x%2B10 and g%28x%29=2x%5E2%2B2x%2B11%2F4 are and respectively.


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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 x%5B1%5D=3%2F2 and y%5B1%5D=49%2F4.
Also, is the second point . So this means that x%5B2%5D=-1%2F2 and y%5B2%5D=9%2F4.



d=sqrt%28%28x%5B1%5D-x%5B2%5D%29%5E2%2B%28y%5B1%5D-y%5B2%5D%29%5E2%29 Start with the distance formula.


d=sqrt%28%283%2F2--1%2F2%29%5E2%2B%2849%2F4-9%2F4%29%5E2%29 Plug in x%5B1%5D=3%2F2, x%5B2%5D=-1%2F2, y%5B1%5D=49%2F4, and y%5B2%5D=9%2F4.


d=sqrt%28%282%29%5E2%2B%2849%2F4-9%2F4%29%5E2%29 Subtract -1%2F2 from 3%2F2 to get 3%2F2--1%2F2=4%2F2=2.


d=sqrt%28%282%29%5E2%2B%2810%29%5E2%29 Subtract 9%2F4 from 49%2F4 to get 49%2F4-9%2F4=40%2F4=10.


d=sqrt%284%2B%2810%29%5E2%29 Square 2 to get 4.


d=sqrt%284%2B100%29 Square 10 to get 100.


d=sqrt%28104%29 Add 4 to 100 to get 104.


d=2%2Asqrt%2826%29 Simplify the square root.


So our answer is d=2%2Asqrt%2826%29


So the exact distance between the two vertices is 2%2Asqrt%2826%29 units.