The function f is defined as follows. f(x) = x^2 +2 If the graph of f is translated vertically upward by 6 units, it becomes the graph of a function h. Find the expression for h(x) h(x) = ?

Question

The function f is defined as follows.

f(x) = x^2 +2

If the graph of f is translated vertically upward by 6 units, it becomes the graph of a function h.

Find the expression for h(x)

h(x) = ?

Please help me solve this question!!!

Answer 1

Translating a function vertically (along the y-axis) always involves adding a constant c to the original function. More specifically, let's say you have an original function f(x).

A vertical translation of this function is expressed as f(x) + c, where c is the amount you are translating.

If c >0, you are translating upwards.

If c<0, you are translating downwards.

So let's apply this to the function you have -- f(x) = x² +2.

If you are translating upwards by 6 units, then c = 6.

Write out the expression f(x) + c, where f(x) is x² +2 and c = 6. Simply and that's your answer.

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Answer 2

Step 1:

First, lets understand the translation for a graph of a function, f(x), in an (x,y) coordinate system:

Vertical Translation is the up/down movement of the function, f(x), along the y-axis to achieve a new function, h(x). Since f(x) = y, we recognize that we can increase y for every x-value by adding/subtracting a constant value, C, to the overall function:

h(x) = f(x) ± C (1)

A horizontal translation is the left/right movement of a function, f(x), along the x-axis to achieve a new function, h(x). This is not asked for in this problem, but we recognize that when translating a function horizontally, the y-value does not change. By adding/subtracting a value constant k to the x-value within the function, the function will have to shift along the x-axis to meet the y-value in f(x):

h(x) f(x ± k) (2)

Step 2:

Let's determine what is asked for and translate the function f(x) accordingly to achieve the new function, h(x):

f(x) = x² + 2

h(x) = f(x), shifted vertically upward by 6 units

To translate f(x) vertically, we will set our constant value, C = 6. Note this is an upward shift, therefore C is positive. Lets solve for h(x) in equation 1:

h(x) = f(x) ± C

h(x) = x² + 2 + 6

h(x) = x² + 8

Notes:

When translating the graph of a function horizontally and/or vertically, the following is true:

a) The shape of the function shall remain the same (it is not being scaled).

b) The orientation of the function shall remain the same (it is not being rotated or reflected).

c) The function may be translated both horizontally and vertically, should the exam problem ask for it, the translated function will follow the formula:

h(x) = f(x ± k) ± C (3)

Answer 3

To solve this problem, you need to understand how vertical translations affect the graph of a function.

The function f(x) = x² + 2 is a parabola that opens upwards with a vertical shift of 2 units.

When the graph of a function is translated vertically upwards by 6 units, you simply add 6 to the original function.

Steps:

1. The original function is:
2. f(x)=x2+2f(x) = x^2 + 2f(x)=x2+2
3. To translate the graph upward by 6 units, you add 6 to the entire function: h(x)=f(x)+6h(x) = f(x) + 6h(x)=f(x)+6
4. Substitute the expression for f(x)f(x)f(x) into this equation: h(x)=(x2+2)+6h(x) = (x^2 + 2) + 6h(x)=(x2+2)+6
5. Simplify the expression: h(x)=x2+8h(x) = x^2 + 8h(x)=x2+8

Final Answer:

The expression for h(x)h(x)h(x) is:

h(x)=x2+8h(x) = x^2 + 8h(x)=x2+8

Answer 4

Translation up and down is added outside of what happens to x: f(x) + k. They go with sign. + is up, - is down. f(x) + 6 translates 6 units up.

Translation right and left happen inside what is done to x: f(x-h). They are opposite of sign - left is positive, right negative. f(x+6) translates 6 units left.

x^2 + 2 + 6 = x^2 + 8 = h(x)

Answer 5

We want to think about how to change a function to move left and right, or up and down. First, think about what the root function looks like. g(x) = x² is our root function, which should look like a parabola opening upward, with its vertex at (0,0). Now let's consider the function we have:

f(x) = x² + 2

How is this different from the root function?

Has this function shifted from the root function?

How can you be sure?

Enter 0 for x, to find out the y-intercept. You should find this is now at (0, 2). How has this shifted from the base function?

So what we have done here is we have actually changed g(x) = x² + 0 to f(x) = x² + 2, which shifted the function vertically upward by 2.

What can you change now to increase it vertically by 6 additional units?

Now be careful! ACT and SAT both like to put trap answers, so we should expect an answer that looks like

x² + 6

Can you tell me how many units this was shifted from f(x)?

Now that you have a handle on shifting functions vertically, how would you shift this 6 units to the right?

Remember, start with the base function, and try to shift that. If we EXPECT that the vertex is supposed to be at the point (6, 0), we can plug in x = 6 to our new function, and should find h(6) = 0. If you got h(6) = 12, then you added inside the parentheses.

Remember when shifting graphs, inside the parentheses, we shift with the opposite sign. When shifting vertically, it stays with the same sign.

If you've solved this correctly, now, you should have

h(x) = (x - 6)²

What if there are no answers that look like this?

You'll need to expand the binomial above!