Transformations of Parabolic Functions - Algebra 2

The function shifts a function f(x) units to the left. Conversely, shifts a function f(x) units to the right. In this question, we are translating the graph two units to the left.

To translate along the y-axis, we use the function or .

The portion results in the graph being shifted 3 units to the left, while the results in the graph being shifted six units down. Vertical shifts are the same sign as the number outside the parentheses, while horizontal shifts are the OPPOSITE direction as the sign inside the parentheses, associated with .

The parent function of a parabola is where are the vertex.

The original graph of a parabolic (quadratic) function has a vertex at (0,0) and shifts left or right by h units and up or down by k units.

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This function then shifts 1 unit left, and 4 units down, and the negative in front of the squared term denotes a rotation over the x-axis.

Correct Answer:

Without doing much work or manipulation of the function, we can use our knowledge of Vertex Form of quadratic functions, which is

with being the coordinates of the vertex. Knowing this, we can analyze our function to find the vertex... vertex: .

Note: This function is simply a transformation of the function

When transforming paraboloas, to translate up, add to the equation (or add to the Y).

To translate to the left, add to the X.

Don't forget that if you add to the X, then since X is squared, the addition to X must also be squared.

with the shift up 5 becomes: .

Now adding the shift to the left we get:

.

To move unit down, subtract from Y (or from the entire equation) , so subtract 1.

To move unit to the left, add to X (don't forget, that since you are squaring X, you must square the addition as well).

With the move down our equation becomes: .

Now to move it to the left we get .

The parent function for a parabolic function is where is the center of the parabola. To shift the parabola left of right, the value of h changes. Since there is a negative sign in the parent function, a positive value moves the parabola to the left and a negative value moves it to the right.

Translations that effect x must be directly connected to x in the function and must also change the sign. So when the function was translated right two spaces, a must be connected to the x value in the function.

Translation that effect y must be directly connected to the constant in the funtion - so when the function was translated up 4 spaces a +4 must be added to the (-5) in the original function.

When both of these happen in the function the new function must become:

Because the parent function is , we can write the general form as:

.

a is the compression or stretch factor.

If , the function compresses or "narrows" by a factor of a.

If , the function stretches or "widens" by a factor of a.

b represents how the function shifts horizontally.

If b is negative, the function shifts to the left b units.

If b is positive, the function shifts to the right b units.

c represents how the function shifts vertically.

If c is positive, the function shifts up c units.

If c is negative, the function shifts down c units.

For our problem, a=3, b=-2, and c=5. (Remember that even though b is negative, the negative from the "general form" makes the sign positive). It follows that we have a compression by a factor of 3, a horizontal shift to the left 2 units, and a vertical shift up 5 units.

Right now, the vertex for this point is , so to shift it up 4 and left 5 would place it at .

This gives us an equation of .

We could also get this by adding 5 inside the parentheses for the left 5 shift, and adding 4 on the outside for the up 4 shift.

Flipping this equation over the x-axis means that the sign of y changes.

The easiest way to accomplish this is just to multiply everything by -1.

To flip this over the -axis, the sign of x changes.

This entails changing to .

.

Perhaps a simpler way to think about this is that the vertex for this parabola is at .

If we flip the equation over the y-axis, it will place the vertex at , making our new equation .

To solve this problem, we could complete the square and shift the equation that way, but the vertex ends up being so this may not be an ideal method. Instead, we know we're shifting the equation 3 units to the right, so we can just plug in for every appearance of x:

To simplify, first expand

Now we can plug that in and continue simplifying:

distribute the 2; combine -3 and -5

combine like terms -12x and x; 18 and -8

Now we want to flip this over the x-axis, meaning that the y coordinates change sign.

This means we have to multiply everything by -1, or simply change the sign of every term on the right side:

First, reflect the equation over the y-axis by switching the sign of x:

Now shift up 2 by adding 2:

Now shift left 4 by adding 4 to x:

first expand

Now multiply

Now plug those back in:

combine like terms

To shift up 6 units, just add 6:

To shift to the right 3, subtract 3 from x:

First expand :

now this gives us:

distribute the 2 and the 4:

combine like terms:

Below is the standard equation for parabolas;

Therefore,

and

thus,

the translation from the parent function is down three units, right one unit.

Shifting up and down will result in a change in the y-intercept.

Add four to the equation.

Shifting the parabola to the left two units will change the inner term to , which will be .

Replace the quantity with .

The new equation is:

Vertical shifts will change the value of the y-intercept. Since this function is to be shifted up two units, add two to the right side of the equation.

This graph shifted two units to the left indicates that its zeros will also shift to the left two units, which means that the term will become .

Rewrite the equation and expand the binomials.

The new equation is:

Shifting this graph three units to the right means that the x-variable will need to be replaced with . Rewrite the equation.

Use the FOIL method to simplify the binomial.

Simplify the right side.

The equation becomes:

The answer is: