Question

PLEASE HELP!! what is the equation for the parabolic function that goes through the points (-3, 67) (-1, 1) and has a stretch factor of 9

Answer 1

AEUJNDUOMBDWEIOOMXAIKMOOESVNOUEWDCYVUJOLDS

Answer 2

`f(x)=9(x+(1)/(6))^2-(21)/(4)`

Explanation:

So, we need to find the equation of a parabolic function that goes through the points (-3,67) and (-1,1) and has a stretch factor of 9.

In other words, we want to find a quadratic with a vertical stretch of 9 that goes through the points (-3,67) and (-1,1).

To do so, we first need to write some equations. Let's use the vertex form of the quadratic equation. The vertex form is:

`f(x)=a(x-h)^2+k`

Where a is the leading coefficient and (h,k) is the vertex.

Since a is the leading coefficient, it's also our stretch factor. Thus, let a equal 9.

Also, we have two points. We can interpret them as functions. In other words, (-3,67) means that f(-3) equals 67 and (-1,1) means that f(-1) equals 1. Write the two equations:

`f(x)=a(x-h)^2+k\\f(-3)=67=9((-3)-h)^2+k`

And:

`f(x)=a(x-h)^2+k\\f(-1)=1=9((-1)-h)^2+k`

Now, we essentially have a system of equations. Thus, to find the original equation, we just need to solve for the vertex. To do so, first isolate the k term in the second equation:

`1=9(-1-h)^2+k\\k=1-9(-1-h)^2`

Now, substitute this value to the first equation:

`67=9(-3-h)^2+k\\67=9(-3-h)^2+(1-9(-1-h)^2)`

And now, we just have to simplify.

First, from each of the square, factor out a negative 1:

`67=9((-1)(h+3))^2+(1-9(((-1)(h+1))^2)`

Power of a product property:

`67=9((-1)^2(h+3)^2)+(1-9((-1)^2(h+1)^2))`

The square of -1 is positive 1. Thus, we can ignore them:

`67=9(h+3)^2+(1-9(h+1)^2)`

Square them. Use the trinomial pattern:

`67=9(h^2+6h+9)+(1-9(h^2+2h+1))`

Distribute:

`67=(9h^2+54h+81)+(1-9h^2-18h-9)`

Combine like terms:

`67=(9h^2-9h^2)+(54h-18h)+(81+1-9)`

The first set cancels. Simplify the second and third:

`67=36h+73`

Subtract 73 from both sides. The right cancels:

`67-73=36h+73-73\\36h=-6`

Divide both sides by 36:

`(36h)/36=(-6)/36\\h=-1/6`

Therefore, h is -1/6.

Now, plug this back into the equation we isolated to solve for k:

`k=1-9(-1-h)^2`

First, remove the negative by simplifying:

`k=1-9((-1)(h+1))^2\\k=1-9((-1)^2(h+1)^2)\\k=1-9(h+1)^2`

Plug in -1/6 for h:

`k=1-9(-(1)/(6)+1)^2`

Add. Make 1 into 6/6:

`k=1-9(-(1)/(6)+(6)/(6))^2\\ k=1-9((5)/(6))^2`

Square:

`k=1-9((25)/(36))`

Multiply. Note that 36 is 9 times 4:

`k=1-9((25)/(9\cdot4))\\ k=1-(25)/(4)`

Convert 1 into 4/4 and subtract:

`k=(4)/(4)-(25)/(4)\\ k=-(21)/(4)`

So, the vertex is (-1/6, -21/4).

Now, plug everything back into the very original equation with 9 as a:

`f(x)=a(x-h)^2+k\\f(x)=9(x+(1)/(6))^2-(21)/(4)`

And this is our answer :)