Use distribution to simplify the equation.

The equation in slope-intercept form is:  

Shift the equation down six units means subtract the y-intercept by six.

The answer is:  

Use the given function  to find the linear transformation defined by .  

First understand that in order to write  we have to take our function  and evaluate it for  and then add  to the result. 

Adding , 

Now that we have , compute the inverse . Conventionally we replace the  notation with the  notation and solve for 

Therefore, 

At this point it's conventional to interchange  and  to write the inverse since we want to express it as a function of 

Therefore the slope is  and the y-intercept is 

In order to take an action from standard form to slope-intercept form you want to make it of the form:

where  is the y-intercept (constant/number without a variable attached to it)

 is the slope and coefficient of the  term

and the equation is set equal to  making it easier to plot graphically.

Given:

I. Isolate  on one side of the equation. This is done by shifting either  over to the other side of equation or the  term and the constant to the other side of the equation. It is generally preferable to shift it to make so the  term is positive by itself to simply operations and sign mixups. So in this case both  and  would be subtracted from both sides of the equation leaving:

II. Now that  is isolated by itself you want to simplify the equation so there is no coefficient other than  attached to , so in this case it'd mean dividing both sides by   leaving:

III. Simplify the expression. If you are given fractions that are divisible by each other they can be simplified.

In the case   is simplified to  and   is simplified to  leaving the final answer of:

The vertex form of a quadratic equation is given by

Where the vertex is located at

giving us .

To find the x-intercepts of the equation, we set the numerator equal to zero.

We can determine the lowest possible value of the expression by finding the -coordinate of the vertex of the parabola graphed from the equation . This is done by rewriting the equation in vertex form.

The vertex of the parabola  is the point .

The parabola is concave upward (its quadratic coefficient is positive), so  represents the minimum value of . This is our answer.

Vertex of quadratic equation  is given by .

For ,

,

so the coordinate of vertex is .

Assume y=0,

 , 

In order to find the vertex of a parabola, our first step is to find the x-coordinate of its center. If the equation of a parabola has the following form:

Then the x-coordinate of its center is given by the following formula:

For the parabola described in the problem, a=-2 and b=-12, so our center is at:

Now that we know the x-coordinate of the parabola's center, we can simply plug this value into the function to find the y-coordinate of the vertex:

So the vertex of the parabola given in the problem is at the point

This is a quadratic function. The -coordinate of the vertex of the parabola can be determined using the formula , setting :

Now evaluate the function at :