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3 years ago

What is the vertex form of y=x^2-6x+6

Mathematics

2 answers:

soldier1979 [14.2K]3 years ago

7 0

$y=x^2-6x+6\\y=x^2-6x+9-3\\y=(x-3)^2-3$

andrezito [222]3 years ago

3 0

Answer:

y = (x - 3)² - 3

Step-by-step explanation:

The equation of a parabola in vertex form is

y = a(x - h)² + k

where (h, k) are the coordinates of the vertex and a is a multiplier

To obtain this form use the method of completing the square

add/subtract (half the coefficient of the x- term )² to x² - 6x

Given

y = x² - 6x + 6

= x² + 2(- 3)x + 9 - 9 + 6

= (x - 3)² - 3 ← in vertex form

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Step-by-step explanation:

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3 years ago

Read 2 more answers

A ball is thrown into the air from a height of 4 feet at time t = 0. The function that models this situation is h(t) = -16t2 + 6

galina1969 [7]

<h2>Hello!</h2>

The answers are:

a) The height of the ball after 3 seconds is 49 feet.

b) The maximum height of the ball is 66 feet.

c) That the ball hit the ground after 4 seconds.

d) The domain would be only the positive real numbers, from 0 to 4, since we found that the ball hit the ground at t equal to 4 seconds. However, if we were talking about a quadratic function with no time involved, the domain would be all the real numbers.

Since we are working with a quadratic function which describes the ball's motion in function of the time, we need to remember the following:

- The general equation of the parabola is:

$y=ax^{2} +bx+c$

- We can calculate the coordinates of the vertex of the parabola using the following formula:

$x_{vertex}=\frac{-b}{2a}$

- Evaluating a function means replacing the variable with the given value to evaluate.

The given function is:

$h(t)=-16x^{2}+63t+4$

Where,

$a=-16\\b=63\\c=4$

Now, calculating we have:

a) What is the height of the ball after 3 seconds?

We need to evaluate the time of 3 seconds into the function, so:

$h(3)=-16(3)^{2} +63(3)+4=49feet$

So, the height of the ball after 3 seconds is 49 feet.

b) What is the maximum height of the ball?

Since the function is describing the motion of a ball thrown into the air, we can find the maximum height by finding the y-coordinate of the vertex. If the parabola opens downward or upward, the vertex will be always the highest or the lowest point of the parabola.

So, calculating the vertex we have:

$x_{vertex}=\frac{-b}{2a}=\frac{-63}{2*-16}\\\\x_{vertex}=\frac{-63}{2*-16}=\frac{-63}{-32}=1.97$

Then, replacing "x" into the equation of the parabola, we find the y-coordinate of the vertex:

$y=-16(1.97)^{2}+63(1.97)+4=-16*3.88+63*1.97+4\\y=-16*3.88+63*1.97+4=-62.08+124.11+4=66.03$

So, if the y-coordinate is 66.03, the maximum height of the ball is 66.03 feet, or 66 feet (rounded to the nearest foot).

c) When will the ball hit the ground?

We can find the time when the ball hit the ground by making equal to 0 the function and finding the roots (zeroes)

Since it's a quadratic function, we can find the zeroes using the quadratic equation:

$\frac{-b+-\sqrt{b^{2}-4ac } }{2a}$

Substituting a, b and c, we have:

$\frac{-b+-\sqrt{b^{2}-4ac } }{2a}=\frac{-63+-\sqrt{63^{2}-4*(-16)*(4)} }{2*(-16)}\\\\\frac{-63+-\sqrt{63^{2}-4*(-16)*(4)} }{2*(-16)}=\frac{-63+-\sqrt{3969+256} }{-32}\\\\\frac{-63+-\sqrt{3969+256} }{-32}=\frac{-63+-\sqrt{4225} }{-32}=\frac{-63+-(65)}{-32}\\\\t1=\frac{-63-(65)}{-32}=4\\\\t1=\frac{-63+(65)}{-32}=-0.06$

Now, since negative time does not exists, we can conclude that the ball hit the ground after 4 seconds.

d) what domain makes sense for the function?

Since the function represents the motion of the thrown ball at "t" time, the domain would be only the positive real numbers, from 0 to 4, since we found that the ball hit the ground at t equal to 4 seconds. However, if we were talking about a quadratic function with no time involved, the domain would be all the real numbers,

Note: I've attached the graph of the function.

Have a nice day!

5 0

3 years ago

samantha threw an apple out of a window from the tenth floor of her appartment building. the equation y = -16t^2+120 can be used

ella [17]

Answer:
time taken = 2.74 seconds

Explanation:
We are given that:
y = -16t² + 120 models the situation where:
y is the floor number
t is the time taken for the apple to hit the ground

We are also given that the floor number (y) is 10. Therefore, all we have to do is substitute with y in the above equation and solve for the time t as follows:
y = -16t² + 120
10 = -16t² + 120
120 - 10 = 16t²
110 = 16t²
t² = 110/16
t² = 6.875
either t = 2.74 seconds ..........> accepted
or t = -2.74 seconds .........> rejected as time cannot be in negative

Hope this helps :)

5 0

3 years ago

Steve earns $9.45 less than twice the amount Bill earns. Steve earns $152.55. How much does Bill earn? Write an equation, Use x

Anna11 [10]

Equation
2x-9.45=152.55
x=81

5 0

3 years ago

Use inductive reasoning to predict the next line in the sequence of computations.

USPshnik [31]

Answer:

1 + (1+6) + (1+6+6) + (1+6+6+6) = 4 × n

Step-by-step explanation:

The computation of the next line in the sequence is shown below:

As we can see that there is an increase of 6 in the left side and on the right hand side there is an increase of one

So,

1 = 1 × n

1 + (1+6) = 2 × n

1 + (1+6) + (1+6+6) = 3 × n

So, the next line would be

1 + (1+6) + (1+6+6) + (1+6+6+6) = 4 × n

i.e.

1 + 7 + 13 + 19 = 4 × 10

As 1 + 7 + 13 + 19 = 40

And 40 = 4 × 10

5 0

3 years ago