Write an equation in slope-intercept form of a line with the following characteristics: parallel to the graph of: -4x + 3y= 12 and passes through the origin.

Answers

Answer 1

Answer: The line that's already there: -4x + 3y= 12

Add 4x to each side: 3y = 4x + 12

Divide each side by 3 : y = 4/3 x + 4

The line that's already there has a slope of 4/3.
Any line parallel to it has the same slope but different y-intercept.
If the new line passes through the origin, then its y-intercept is zero.

The equation of the new line is y = 4/3 x

Answer 2

Answer: Slope of given line is m=-a/b =4/3

So searched equation is:

y-0=4/3(x-0)
y = 4/3x

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Bob and mark talk about their families. Bob says he has 3 kids, the product of their age is 72. He gives another clue: the sum of the ages of his children. Mark points out there is still not enough information to accurately guess. Finally, bob says, my youngest child called justice .
Mark can then correctly determine the ages of bobs children .
What are the ages ?

Answers

Bob says he has 3 kids, the product of their age is 72.
So the possible ages of Bob's kids can be taken from the factors of 72

72 = 2 * 2 * 2 * 3 * 3

Finally, bob says, my youngest child called justice.
So meaning, Bob has a youngest child. He didn't mention about a twin.

So the possible ages the following:
1, 2, 36 --> sum: 39
1, 4, 18 --> sum: 23
1, 8, 9 --> sum: 18

Once the sum of the ages will be revealed, then the possible ages are mentioned above.

The geometric period was mostly characterized by

Answers

The geometric period was mostly characterized by diversity.

What did the sea monster say after eating 27 ships carrying potatoes?

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"I should stop setting up for bad jokes..."

or "Fishin' ships are pretty good..."

Find x
(4x - 3) – ( x + 5) = 3(10 - x)
please.....

Answers

Answer: 19/3

Step-by-step explanation:

$4x+3-x-5=30-3x\n\n3x-8=30-3x\n\n6x-8=30\n\n6x=38\n\nx=(19)/(3)$

When looking at a function graph, which value determines whether a graphed relation is a function or not? Domain or Range? Explain why.

Answers

It would be the domain, you can only have a unique y-value for each x, this is called the vertical line test. You can have the same y values for other points of y but not for the same point of x.

The height y of a ball (in feet) is given by the function and x is the horizontal distance traveled by the ball.a)How high is the ball when it leaves the child’s hand?
"b) How high is the ball at its maximum heigh
c) Explain in words the method you used in part b.
d) What is the horizontal distance traveled by the ball when it hits the ground?
e) Explain in words what you did to find your answer for part d.

Answers

Given: The height y of a ball (in feet) is given by the function y=-1/12x^2+2x+4 and x is the horizontal distance traveled by the ball.

Part A: How high is the ball when it leaves the child's hand?

Right after the ball leaves the child's hand, it has travelled 0 feet horizontally. Horizontal distance is represented by x, so we could say that x = 0.
Plug in 0 for our equation and solve for y, the height.

$y=-(1)/(12)x^2+2x+4\n\ny=(1)/(12)\cdot0^2+2\cdot0+4\n\ny=0+0+4\n\n\boxed{y=4}$

Part B & C: How high is the ball at its maximum height?

What we basically want to do is find the vertex of the function.
There are multiple ways to do this. You could graph it or make a table, but this method is not efficient.
The method I am going to go over right now is putting the equation in vertex form.

$y=-(1)/(12)x^2+2x+4$

Move the constant to the left side.

$y-4=-(1)/(12)x^2+2x$

Factor out the x² coefficient.

$y-4=-(1)/(12)(x^2-24x)$

Find out which number to add to create a perfect square trinomial.
(Half of 24 is 12, 12 squared is 144. We have to add 144/-12 (which is -12) to each side so that we end up with 144 inside the parentheses on the right side)

$y-4-12=-(1)/(12)(x^2-24x+144)$

Factor the perfect square trinomial and simplify the right side.

$y-16=-(1)/(12)(x-12)^2$

Isolate y on the left side.

$y=-(1)/(12)(x+12)^2+16$

And now we are in vertex form.
Vertex form is defined as y = a(x-h)² + k with vertex (h, k).
In this case, our vertex is (12, 16).

You could've also taken the shortcut that for any quadratic f(x) = ax² + bx + c, the vertex (h, k) is (-b/2a, f(h)). That's basically a summation of this method which you can use if your teacher has taught it to you.

Part D & E: What is the horizontal distance travelled by the ball when it hits the ground?
When the ball hits the ground, y is going to be 0, since y is the ball's height.
There are many ways to solve a quadratic...split the middle, complete the square, and the quadratic formula.

$-(1)/(12)x^2+2x+4=0$

Solving by splitting the midlde
If your quadratic has fractions, this is not a good option.

Solving by completing the square
Move the constant over the right side.

$y=-(1)/(12)x^2+2x=-4$

Divide by the x² coefficient.
(Dividing by -1/12 is the same as multiplying by its reciprocal, -12.)

$x^2-24x=-4*-12$

Simplify the right side.

$x^2-24x=48$

Halve the x coefficient, square it, and then add it to each side.
(Half of -24 is -12, and -12 squared is 144.)

$x^2-24x+144=192$

Factor the perfect square trinomial.

$(x-12)^2=192$

Take the square root of each side.

$x-12=\pm√(192)$

192 = 8 × 8 × 3, so we can simplify √192 to 8√3.
Add 12 to each side and we get our answer.

$x=12\pm8√(3)$

Our function does not apply when x or y is less than 0, of course.
12-8√3 is negative, so this cannot be our answer.
So, the ball had travelled 12+8√3 feet at the time when it hit the ground.

Solving with the quadratic formula
For any equation ax² + bx + c = 0, the solution for x is $(-b\pm√(b^2-4ac))/(2a)$ .

Our equation, y=-1/12x^2+2x+4, has a = -1/12, b=2, and c=4.
Let's plug these values into the quadratic formula.

$\frac{-2\pm\sqrt{2^2-4\cdot(-1)/(12)\cdot4}}{2\cdot(-1)/(12)}=\frac{-2\pm\sqrt{4-\frac{-4}3}}{\frac{-1}6}=\frac{-2\pm\sqrt{(16)/(3)}}{\frac{-1}6}=\frac{-2\pm(4)/(√(3))}{\frac{-1}6}$

Dividing by a fraction is the same as multiplying by its reciprocal...

$-6(-2\pm(4)/(√(3)))=12\pm(-24)/(√(3))=12\pm(24)/(√(3))=12\pm\frac{24√(3)}3=\boxed{12\pm8√(3)}$

Of course, we only want the positive value, 12+8√3.

Revisiting Part B & C:
Since parabolae are symmetrical, if you know two values of x for some value of y (like the x-intercepts we just found in part B) then you can find the average between them to find what the x value of the vertex is, then plug that in to find the y value of the vertex (the height we want)

The average between 12+8√3 and 12-8√3 is 12. Plug that in and we get 16!

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