Find the Vertex y=-2x^2-16x-24

Solution:

Step:1

Transform the quadratic equation into the vertex form.

Step:1.1

Complete the square for the quadratic expression $-2x^2-16x-24$.

Step:1.1.1

Identify the coefficients $a$, $b$, and $c$ from the standard form $a{x}^2+bx+c$.

$$a=-2,b=-16,c=-24$$

Step:1.1.2

Recall the vertex form of a quadratic equation: $a(x+d{)}^2+e$.

Step:1.1.3

Calculate the value of $d$ using the formula $d=-\frac{b}{2a}$.

Step:1.1.3.1

Insert the known values of $a$ and $b$ into the formula.

$$d=-\frac{-16}{2\cdot -2}$$

Step:1.1.3.2

Perform the simplification of the expression.

Step:1.1.3.2.1

Extract the common factor from the numerator.

$$d=-\frac{2\cdot -8}{2\cdot -2}$$

Step:1.1.3.2.2

Eliminate the common factors.

$$d=-\frac{\cancel{2}\cdot -8}{\cancel{2}\cdot -2}$$

Step:1.1.3.2.3

Rewrite the simplified expression.

$$d=-\frac{-8}{-2}=4$$

Step:1.1.4

Determine the value of $e$ using the formula $e=c-a(d^2)$.

Step:1.1.4.1

Plug the values of $a$, $b$, and $c$ into the formula.

$$e=-24-(-2)(4^2)$$

Step:1.1.4.2

Simplify the expression.

Step:1.1.4.2.1

Calculate each term independently.

Step:1.1.4.2.1.1

Compute the square of $4$.

$$e=-24-(-2)(16)$$

Step:1.1.4.2.1.2

Multiply $-2$ by $16$.

$$e=-24+32$$

Step:1.1.4.2.1.3

Combine the terms.

$$e=8$$

Step:1.1.5

Insert the values of $a$, $d$, and $e$ into the vertex form equation.

$$y=-2(x+4{)}^2+8$$

Step:1.2

Set $y$ to the new expression.

$$y=-2(x+4{)}^2+8$$

Step:2

Identify the vertex form coefficients $a$, $h$, and $k$.

$$a=-2,h=-4,k=8$$

Step:3

Locate the vertex $(h,k)$.

$$(h,k)=(-4,8)$$

Step:4

The vertex of the parabola is $(-4,8)$.

Knowledge Notes:

To find the vertex of a parabola given in the standard form $y=a{x}^2+bx+c$, one can use the process of completing the square to rewrite the equation in vertex form, which is $y=a(x-h{)}^2+k$, where $(h,k)$ is the vertex of the parabola.

• Completing the Square: This involves creating a perfect square trinomial from the quadratic equation, which then can be factored into the form $(x+d{)}^2$. This process often requires adding and subtracting the same value inside the equation to maintain equality.

• Vertex Form of a Parabola: The vertex form is useful because it clearly shows the vertex $(h,k)$ of the parabola. The coefficient $a$ affects the direction and width of the parabola, $h$ affects the horizontal position, and $k$ affects the vertical position.

• Vertex Calculation: The vertex can be found using the formulas $h=-\frac{b}{2a}$ and $k=c-a(h^2)$. These formulas come from the process of completing the square and represent the coordinates of the vertex in terms of the coefficients $a$, $b$, and $c$ from the standard form of the quadratic equation.

• Sign of Coefficient $a$: The sign of $a$ determines whether the parabola opens upwards ($a>0$) or downwards ($a<0$). In this case, since $a=-2$, the parabola opens downwards.

• Simplification: When simplifying expressions, it's important to carefully perform operations such as factoring, canceling common factors, and combining like terms to arrive at the correct values for $d$ and $e$.