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The greatest common factor of 32 and 52 is: 4

Factors of 32: 1, 2, 4, 8, 16, 32
Factors of 52: 1, 2, 4, 13, 26

The common factors between them are 1, 2, 4, but the greatest common factor between them is 4.

7 0

3 years ago

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a road follows the shape of a parabola f(x)=3x2– 24x + 39. A road that follows the function g(x) = 3x – 15 must cross the stream

Nonamiya [84]

the coordinates where the bridges must be built is $(3,-6)$ and $(6,3)$ .

Step-by-step explanation:

Here we have , a road follows the shape of a parabola f(x)=3x2– 24x + 39. A road that follows the function g(x) = 3x – 15 must cross the stream at point A and then again at point B. Bridges must be built at those points.We need to find Identify the coordinates where the bridges must be built. Let's find out:

Basically we need to find values of x for which f(x) = g(x) :

⇒ $f(x)-g(x)=0$

⇒ $3x^2- 24x + 39-(3x - 15 ) =0$

⇒ $3x^2- 24x + 39-3x + 15 =0$

⇒ $3x^2- 27x + 54 =0$

⇒ $x^2- 9x + 18 =0$

⇒ $x^2- 6x-3x + 18 =0$

⇒ $x(x- 6)-3(x - 6) =0$

⇒ $(x-3)(x- 6) =0$

⇒ $x=3 , x=6$

Value of g(x) at x = 3 : y=3x -15 = 3(3)-15 = -6

Value of g(x) at x = 6 : y=3x -15 = 3(6)-15 = 3

Therefore , the coordinates where the bridges must be built is $(3,-6)$ and $(6,3)$ .

7 0

3 years ago

Compare the two graphs and explain the transformation that was applied to f(x) in order to look exactly like the graph of g(x).

Neporo4naja [7]

The two graphs are represented below.

Answer and Step-by-step explanation: One graph can "transform" into another through changes in the function.

There are 3 ways to change a function:

Shifting: it adds or subtracts a constant to one of the coordinates, thus changing the graph's location. When the y-coordinate is added or subtract and the x-coordinate is unchanged, there is a vertical shift. If it is the x-coordinate which changes and y-coordinate is kept the same, the shift is a horizontal shift;
Scaling: it multiplies or divides one of the coordinates by a constant, thus changing position and appearance of the graph. If the y-coordinate is multiplied or divided by a constant but x-coordinate is the same, it is a vertical scaling. If the x-coordinate is changed by a constant and y-coordinate is not, it is a horizontal scaling;
Reflecting: it's a special case of scaling, where you can multiply a coordinate per its opposite one;

Now, the points for f(x) are:

(-5,0) (0,6) (5,-4) (8,0)

And the points for g(x) are:

(-5,-3) (0,-9) (5,1) (8,-3)

Comparing points:

(-5,0) → (-5,-3)

(0,6) → (0,-9)

(5,-4) → (5,1)

(8,0) → (8,-3)

It can be noted that x-coordinate is kept the same; only y-coordinate is changing so we have a vertical change. Observing the points:

(-5,0-3) → (-5,-3)

(0,6-15) → (0,-9)

(5,-4+5) → (5,1)

(8,0-3) → (8,-3)

Then, the vertical change is a Vertical Shift.

Another observation is that y-coordinate of f(x) is the opposite of g(x). for example: At the second point, y-coordinate of f(x) is 6, while of g(x) is -9. So, this transformation is also a Reflection.

Range of a function is all the values y can assume after substituting the x-values.

Domain of a function is all the values x can assume.

Reflection doesn't change range nor domain of a function. However, vertical or horizontal translations do.

Any vertical translation will change the range of a function and keep domain intact.

Then, for f(x) and g(x):

graph translation domain range

f(x) none [-5,8] [-4,6]

g(x) vertical shift [-5,8] [-9,1]

In conclusion, this transformation (or translation) will affect the range of g(x)

5 0

3 years ago