hectictar

These are both the same question:

Thanks....I messed that up!!

3. What is the amplitude of the function graphed?

amplitude  =  maximum y-value  -  minimum y-value   =   6 - (-2)   =   6 + 2   =   8

1. The function shown in the graph is vertically stretched by a factor of 4 to produce a new graph. Which function represents the new graph?

The graph in the picture has a period of  1  and amplitude of  1  , so it is a graph of  y = sin(2π x)

y  =  sin(2π x)     stretched vertically by a factor of  4  is     y  =  4sin(2π x)

2x + 5y  =  -8          Multiply both sides of this equation by  3.

6x + 15y  =  -24

6x  =  16 + a - 15y          Add  15y  to both sides of this equation.

6x + 15y  =  16 + a

So now our two equations are:

6x + 15y  =  -24

6x + 15y  =  16 + a

The system will have infinitely many solutions when both equations are identical.

So the system will have infinitely many solutions when....

-24  =  16 + a          Subtract  16  from both sides of this equation.

-40  =  a

a  =  -40

Yep! it is  n ≥ 9 

discriminant of first one   =   16² - 4(4)(-9)   =   400   ≠   0

discriminant of second one   =   80² - 4(2)(400)   =   3200   ≠   0

discriminant of third one   =   (-6)² - 4(1)(-9)   =   72   ≠   0

discriminant of fourth one   =   (-12)² - 4(4)(9)   =   0

discrimimant of fifth one   =   (14)² - 4(-1)(49)   =   392   ≠   0

The fourth one is the quadratic with one distinct root.

4x² - 12x + 9  =  0     when

x  =  \(\frac{-(-12)\pm\sqrt{(-12)^2-4(4)(9)}}{2(4)}\ =\ \frac{12\pm0}{8}\ =\ \frac{12}{8}\ =\ \frac32\)

The dot is colored in over the  9  so that means  9  is included. So the answer is one of the last two options.

Do you have any guess about which one it is? I'll tell you if it's right or wrong.

Note that   \(\frac{\text{55 miles}}{\text{1 hour}}\ =\ \frac{\text{55 miles}}{\text{60 minutes}}\)

Let  x  be the number of minutes it takes.

\( \frac{\text{55 miles}}{\text{60 minutes}}\cdot x\text{ minutes}\ =\ 310\text{ miles}\\~\\ x\text{ minutes}\ =\ 310\text{ miles}\cdot\frac {\text{60 minutes}}{\text{55 miles}}\\~\\ x\text{ minutes}\ =\ 310\cdot\frac {60}{55}\text{ minutes}\\~\\ x\text{ minutes}\ =\ \frac{18600}{55}\text{ minutes}\\~\\ x\text{ minutes}\ \approx\ 338\text{ minutes} \\~\\ x\ \approx\ 338\)

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△BCD is a special right triangle called a " 30-60-90 triangle "  because its angles are 30°, 60°, and 90°.

In every 30-60-90 triangle, if the side opposite the 30° angle is  n , then

the side opposite the 60° angle is  n√3  and the side opposite the 90° angle is 2n

In △BCD, the side opposite the 30° is  12 , so

the side opposite the 60° angle is  12√3  and the side opposite the 90° angle is  2(12) .

So.....

BC = 12√3

BD =  2(12)  =  24

height of the triangle  =  12√3  cm

area of triangle  =  ( 1/2 )( base )( height )

area of triangle  =  ( 1/2 )( 12 )( 12√3 )  cm²               I'll let you simplify this.