Respuesta :

Answer:

axis of symmetry is [tex]x=\frac{-9}{2}[/tex].

The ordered pair of the vertex is [tex](\frac{-9}{2},\frac{-145}{4})[/tex].

Step-by-step explanation:

Your function is a quadratic.

Compare [tex]x^2+9x-16[/tex] to [tex]ax^2+bx+c[/tex].

You should see that [tex]a=1,b=9,c=-16[/tex].

The x-coordinate of the vertex or the axis of symmetry since the axis symmetry goes through the vertex can be found by computing [tex]\frac{-b}{2a}[/tex].

So here we go!

The axis of symmetry is [tex]x=\frac{-9}{2(1)}=\frac{-9}{2}[/tex].

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

That is the axis of symmetry is [tex]x=\frac{-9}{2}[/tex].

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

[tex]y=x^2+9x-16[/tex].

[tex]y=(\frac{-9}{2})^2+9(\frac{-9}{2})-16[/tex]

Put into calculator:

[tex]y=\frac{-145}{4}[/tex] when [tex]x=\frac{-9}{2}[/tex]

The ordered pair of the vertex is [tex](\frac{-9}{2},\frac{-145}{4})[/tex].

Supernova

Answer:

Vertex: [tex](h,k)\rightarrow(-4.5,-36.25)[/tex]

Axis of symmetry: [tex]x=-4.5[/tex]

Step-by-step explanation:

Finding the Axis of Symmetry:

First I'll find the axis of symmetry. This formula lets us find the a.o.s: [tex]x=\frac{-b}{2a}[/tex].

In [tex]x^2+9x-16[/tex], 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.

  • [tex]x=\frac{-(9)}{2(1)}[/tex]

Simplify this fraction.

  • [tex]x=\frac{-9}{2} =-4.5[/tex]

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.

  • [tex]y+16=x^2+9x[/tex]

Use the completing the square formula: [tex](\frac{b}{2} )^2[/tex]

  • [tex](\frac{9}{2} )^2=20.25[/tex]

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

  • [tex]y+16+20.25=x^2+9x+20.25[/tex]

Combine like terms.

  • [tex]y+36.25=x^2+9x+20.25[/tex]

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.

  • [tex]y+36.25=(x+4.5)(x+4.5)[/tex]
  • [tex]y+36.25=(x+4.5)^2[/tex]

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

  • [tex]y=(x+4.5)^2-36.25[/tex]

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

Vertex form of a quadratic is:

  • [tex]y=a(x-h)^2+k[/tex]

Compare [tex]y=(x+4.5)^2-36.25[/tex] 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 [tex]y=(x+4.5)^2-36.25[/tex] 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 [tex](-4.5, -36.25)[/tex].

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