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

Step-by-step 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:

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:

And:

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:

Now, substitute this value to the first equation:

And now, we just have to simplify.

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

Power of a product property:

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

Square them. Use the trinomial pattern:

Distribute:

Combine like terms:

The first set cancels. Simplify the second and third:

Subtract 73 from both sides. The right cancels:

Divide both sides by 36:

Therefore, h is -1/6.

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

First, remove the negative by simplifying:

Plug in -1/6 for h:

Add. Make 1 into 6/6:

Square:

Multiply. Note that 36 is 9 times 4:

Convert 1 into 4/4 and subtract:

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

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

And this is our answer :)

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