Question

F(x)=x^2+9x-16


What is vertex

Axis of semetry

Answer 1

axis of symmetry is
`x=(-9)/(2)`.

The ordered pair of the vertex is
`((-9)/(2),(-145)/(4))`.

Explanation:

Your function is a quadratic.

Compare
`x^2+9x-16` to
`ax^2+bx+c`.

You should see that
`a=1,b=9,c=-16`.

The x-coordinate of the vertex or the axis of symmetry since the axis symmetry goes through the vertex can be found by computing
`(-b)/(2a)`.

So here we go!

The axis of symmetry is
`x=(-9)/(2(1))=(-9)/(2)`.

When you write your axis of symmetry be sure to write it as an equation.

That is the axis of symmetry is
`x=(-9)/(2)`.

Now that was also the x-coordinate of your vertex. To find the corresponding y-coordinate of the vertex, plug your value for
`x` into

`y=x^2+9x-16`.

`y=((-9)/(2))^2+9((-9)/(2))-16`

Put into calculator:

`y=(-145)/(4)` when
`x=(-9)/(2)`

The ordered pair of the vertex is
`((-9)/(2),(-145)/(4))`.

Answer 2

Vertex:
`(h,k)\rightarrow(-4.5,-36.25)`

Axis of symmetry:
`x=-4.5`

Explanation:

Finding the Axis of Symmetry:

First I'll find the axis of symmetry. This formula lets us find the a.o.s:
`x=(-b)/(2a)`.

In
`x^2+9x-16`, the values of a, b, and c are:

• a: 1
• b: 9
• c: -16

We only need a and b to find the axis of symmetry. Substitute these values into the formula.

• 
  `x=(-(9))/(2(1))`

Simplify this fraction.

• 
  `x=(-9)/(2) =-4.5`

The axis of symmetry of this quadratic function is x = -4.5.

Finding the Vertex:

Now to find the vertex, we have to take into account that this quadratic is in standard form, making it a little harder. We have to convert this function into vertex form.

Start by changing f(x) to 'y' and adding 16 to both sides.

• 
  `y+16=x^2+9x`

Use the completing the square formula:
`((b)/(2) )^2`

• 
  `((9)/(2) )^2=20.25`

Keep the balance by adding 20.25 on the left side and adding it on the right side of the equation.

• 
  `y+16+20.25=x^2+9x+20.25`

Combine like terms.

• 
  `y+36.25=x^2+9x+20.25`

Factor the right side of the equation. Ask yourself, "What two numbers multiply to 20.25 (c) and add up to 9 (b)?" These two numbers are 4.5 and 4.5. Rewrite the right side with factors.

• 
  `y+36.25=(x+4.5)(x+4.5)`
• 
  `y+36.25=(x+4.5)^2`

Isolate y by subtracting 36.25 from both sides of the equation.

• 
  `y=(x+4.5)^2-36.25`

Now this quadratic function is in vertex form, making it super simple to find the vertex using
`(h, k)`.

Vertex form of a quadratic is:

• 
  `y=a(x-h)^2+k`

Compare
`y=(x+4.5)^2-36.25` with the original vertex form and find where h and k are. Those are the x (h) and y (k) values of the vertex.

Since the original vertex form has x - h, the h value in
`y=(x+4.5)^2-36.25` would be a negative since two negatives make a positive. The k value would stay "normal"---negative would mean it is a negative and positive would mean it is a positive number.

Therefore the h value is -4.5, and the k value is -36.25.

The ordered pair of the vertex is
`(-4.5, -36.25)`.