Question

what is the equation of the axis of symmetry of
$y = - 2{x}^{2} - 5x + 3$
and the coordinates for the vertex, find the y- intercept, find the zeros too.

Answer 1

Answer:

You will have to read to get the 4 main points and the axis of symmetry.

Step-by-step explanation:

Let's get the roots out of the way first. The first thing to try is just factoring it.

The x^2 factors have opposite signs so you get

(x        )(-x     )

The next problem is to deal with the 3. It has only numbers making it up: itself and 1. So we can pud the 3 somewhere, but where.  But we know that the signs on the three must be the same +1 and + 3 or - 1 and - 3. It is a pure guess. on which will work

(x  3)(-x + 1) The minus and plus are just place holders. And finally the 2.

(x + 3)(-2x + 1). Now the only way you can check this is to foil it.

-2x^2 - 6x + x + 3

-2x^2 - 5x + 3

We have the factors. This was just guess work.

Roots

x+3 = 0

x = - 3  is one root.

-2x + 1 = 0

- 2x = -1

x = 1/2

Answer for roots (1/2,0)(-3,0)

============================

Now we go for the vertex.

1. take out - 2 only for the first 2 terms.

y = -2(x^2 + (5/2)x ... ) + 3 The three goes outside the brackets.

2. Take the coefficient inf front of the x. 1/2 it and square the results. Add that inside the brackets

y = -2(x^2 + (5/2)x + (5/4)^2 ) + 3

3. Whatever you do inside the brackets, you have to made an adjustment for after the 3. You also have to take the - 2 into account. These last 2 later.

• (5/4)^2    square 5/4
• (25/16)    This goes inside the brackets
• y = -2[x^2 + (5/2)x + (25/16)] + 3

4. What goes after the 3?

• -2(25/16) = - 25/8
• This is what essentially goes after the three. But should it be minus?
• The answer is no. That was something the - 2 brought in with it. Since the - 2 makes 25/8 minus, we should be putting +25/8 after the three. That will mean we have added nothing to the equation.
• y = - 2 (x^2 + (5/2)x + 25/16) )+ 3 + 25/8  Combine
• y = - 2 (x^2 + (5/2)x + 25/16) ) + 6.125

5. Write the vertex form of the equation.

y = -2(x + 5/4)^2 + 6.126

The vertex = (-5/4, 6.125

====================

The axis of symmetry is

x + 5/4 = 0

x = - 5/4

Y intercept

x = 0

y = -2(0)^2 - 5(0) + 3

y intercept = (0,3)

Graph

Red line with the 4 major points is y = -2x^2 - 5x + 3

Blue line is the axis of symmetry x = -5/4