f(x)=-3(x+1)^2+27

Step-by-step explanation:

Forst, simplify the expression by pulling out -3, but leave 24 out.

=-3(x^2+2x)+24

Then complete the square of the equatio inside the parenthesis (remember to subtract from the outside what you add to the inside times -3).

=-3(x^2+2x+1)+24+3

=-3(x+1)^2+27

The equation is now in vertex form.

from that, we now know that the vertex is (-1,27).

Set this equation equal to 0 to find the x intercepts:

0=-3(x+1)^2+27

-27=-3(x+1)^2

9=(x+1)^2

3 and -3=x+1

x=2 and -4

f(x) = - 2(x - )² -

the equation of a quadratic 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

• the coefficient of the x² term must be 1 , factor out - 2

= - 2(x² - x) - 2

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

= - 2(x² + 2(- )x + - ) - 2

= - 2(x - )² + - 2

= - 2(x - )² -

Discussion

1. Put brackets around the first two terms

y = (-x^2 + 6x) + 5

2. Take out the common factor of -1

y = -(x^2 - 6x) + 5

3. Inside the brackets, take 1/2 of - 6 and square it

y = -(x^2 - 6x + ( - 6 / 2)^2 ) + 5

y = -(x^2 - 6x + (- 3)^2 ) ) + 5

y = -(x^2 - 6x + 9 ) + 5

Note: Step 3 is very long. Make sure you work your way through it

4. You have added 9 inside the brackets. It is actually - 9. So add 9 outside to balance the equation out. This is the key step. Make sure you understand it.

y = - (x^2 - 6x + 9) + 5 + 9

5. Express the brackets as a square.

y = - (x + 3)^2 + 14

Discussion

The equation is now in vertex form. The minus tells you that the equation is a maximum. The maximum is located at ( - 3, 14 )

A graph follows to show the results.

5x^2+15x-2

5(x^2+3x)-2

5(x^2+3x+2.25-2.25)-2

5(x^2+3x+2.25)-13.25

5(x+1.5)^2-13.25

final y=5(x+1.5)^2-13.25

axis of symmetry: (1.5, 13.25)

(x - 3)² = - (y - 14)

The maxima of the function is (3,14).

Step-by-step explanation:

The formula of a quadratic function is given by y = - x² + 6x + 5

We have to write this in the vertex form.

Now, y = - x² + 6x + 5

⇒ y = - (x² - 6x + 9) + 9 + 5

⇒ y - 14 = - (x - 3)²

⇒ (x - 3)² = - (y - 14)

This is the equation of the parabola in vertex form.

This parabola has the vertex at (3,14) and the axis of the parabola is parallel to the negative y-axis.

Therefore, the maxima of the function is at (3,14). (Answer)

The vertex form of the given function is and its axis of symmetry is

Step-by-step explanation: The given quadratic function is

We are to rewrite the above function in vertex form and to determine its axis of symmetry.

We have from equation (i),

So, the given function is a parabola with vertex at the point

Therefore, the axis of symmetry is given by

Thus, the vertex form of the given function is and its axis of symmetry is

5(x^2+3x)-2

5(x^2+3x+2.25-2.25)-2

5(x^2+3x+2.25)-13.25

5(x+1.5)^2-13.25

final y=5(x+1.5)^2-13.25

axis of symmetry: (1.5, 13.25)

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