im confused on this question..iv tired working this out for hours..does any1 here knw how 2 work this out ??

Question

This part of the question concerns the graph of the function
f (x) = 1/3 (x − 2)^2 − 3.

(i) Explain how the parabola that is the graph of f can be obtained from the graph of y = x^2 by using appropriate translations and scalings.

(ii) Using your answer to part (i), or otherwise, write down the
coordinates of the vertex of the parabola.

(iii) Find the x-intercepts and the y-intercept of the parabola.

(iv) What is the image set of the function f ? (You should express your answer in interval notation.)

Answer 1

This graph is in a form known as vertex form.

Vertex form states y=a(x-h)^2+k

This sounds fancy but a is the rate of change (like slope)

h is the horizontal shift.

k is the vertical shift.

Your vertex is found at h,k

In your equation f(x)=1/3(x-2)^2-3

The a value or rate of change is 1/3. This means that all y-values of y=x^2 would be multiplied by 1/3.

Since (x-h) = (x-2)
h=+2
Note: h is always opposite of the number we see next to x.
So if you see x -2,
h is +2. So the horizontal shift is right 2.
On the table of y=x^2 you have already multiplied all of the y-values by 1/3. For h(Horizontal Shift) of 2 you would add 2 to all of the x values on your table.

k is always the value at the end of the equation.
Unlike h it is the same as it appears.
Your equation has -3 on the end so
k=-3 and the vertical shift is down 3.
Now that you have multiplied all of the y values by 1/3 (the a value)
And added 2 to all of the x values (since h=+2)
Because k=-3 you will subtract 3 from all of the y values to perfrom your vertical shift
(remember these values have already been multiplied by 1/3)

At this point you will have the table of your shifted graph.

See how I shift the table of y=x^2 below:

Start with the graph:
y=x^2 ( I show x=-3 to x=3)

(x,y)
(-3,9)
(-2,4)
(-1,1)
(0,0)
(1,1)
(2,4)
(3,9)

Since a=1/3 multiply (scale) each y-value in table above by 1/3
y=1/3x^2

(x,y)
(-3,3)
(-2,4/3)
(-1,1/3)
(0,0)
(1,1/3)
(2,4/3)
(3,3)

Since h=2 add 2 to each x value from table above.
This is called shifting or translating the graph right.
y=1/3(x-2)^2

(x,y)
(-1,3)
(0,4/3)
(1,1/3)
(2,0)
(3,1/3)
(4,4/3)
(5,3)

Since k=-3 subtract 3 from each y value from table above.
This is shifting or translating the graph down.
y=1/3(x-2)^2-3

(x,y)
(-1,0)
(0,-5/3)
(1,-8/3)
(2,-3)
(3,-8/3)
(4,-5/3)
(5,0)


These points are the values of your scaled/shifted graph.

ii.
If done correctly your vertex will be the middle value.
In this case it is (2,-3)

Also your vertex is always (h,k)
Since h=2, k=-3 your vertex is:
(2,-3)

iii.
Roots are always the x intercepts (where y =0)
In the created table we see (-1,0) and (5,0)
Your x intercepts are -1 and 5.

Y intercepts are always where x=0
In the created table we see that at x=0, y=-5/3
-5/3 is your y-intercept.

4) Image set is the domain and range values seen when graphing the function (specifically in a calculator).

If you plot the points of each table above you will see how the graph scales and shifts at each step. Best wishes in your work!

Answer 2

PART 1:
f(x) = 1/3(x - 2)² - 3

Break this down in parts:

If you had (1/3)x², it would look just like x² except each y value is multiplied (scaled) by 1/3. It is scaled by 1/3 in the y direction.

If you had (x - 2)², this would be the same as x² but you have to subtract two from each x value before it would look like the x² graph. Thus the graph is shifted to the *right* two units.

Finally, if you had x² - 3, this would just mean every y value was 3 less than x², so the -3 at the end just drops the graph down 3 units vertically.

Putting this all together for (i)
The parabola is scaled by 1/3, moved 2 to the right, down 3.

PART 2:
The vertex of the x² parabola is (0,0), so our new parabola is over 2 and down 3. That would be (2, -3)

PART 3:
Plug in x = 0 to find the *Y* intercept.
f(x) = 1/3(x - 2)² - 3
f(0) = 1/3(0 - 2)² - 3
f(0) = 1/3(-2)² - 3
f(0) = (1/3)(4) - 3
f(0) = 4/3 - 3
f(0) = 4/3 - 9/3
f(0) = -5/3
y-intercept = (0, -5/3)

Plug in y = 0 to find the *X* intercepts.
0 = 1/3(x - 2)² - 3

Add 3 to both sides:
3 = 1/3(x - 2)²

Multiply both sides by 3 to get rid of the fraction:
9 = (x-2)²

Take the square root of both sides:
±3 = (x-2)
x - 2 = 3 or x - 2 = -3
x = 5 or x = -1

x-intercepts: (5, 0) or (-1, 0)

PART 4:
I assume this just means to express the domain and range in interval notation?

Domain (possible values of x):
-∞ < x < ∞
(-∞, ∞)

Range (possible values of y):
-3 ≤ y < ∞
[-3, ∞)

Answer 3

The -2 next to the x causes a horizontal translation of 2 to the right. The -3 at the end causes a vertical translation 3 down. The 1/3 in frone causes a widening of 3 times in scaling.

The vertex is (2, -3) (2 to the right and 3 down)

X intercepts are when y = 0 [or here, f(x) = 0] so 1/3 (x-2)^ - 3 = 0

1/3 (x-2)^2 = 3
(x-2)^2 = 9
x-2 = +/- 3
x = 3 + 2 or -3 + 2

y-intercepts are when x = 0, just plug in 0 for x and see what you get

Sorry, I've never heard of "image set" unless it means the domain and/or range of the translation, which are: domain = all numbers; range = [-3, infinity)

Answer 4

Lets see...

I) well first you would move the x^2 parabola to the right 2 times.. then multiple all the y values with 1/3. Then you would simply move the parabola down 3 values..and the vertex is given to you because the equation form is already completed the square which is (2,-3)

ii) (2,3)

iii) X int is -1 and 5

Y int > just equal x to zero and this would give you the y int so
f (x) = 1/3 (0 − 2)^2 − 3 =
1.66666667

iv) i didnt understand the question...

anyways i hope this helped :p

Answer 5

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