Find the Vertex y=-2x^2-16x-24
The given problem involves determining the vertex of the quadratic function y = -2x^2 - 16x - 24. The vertex of a quadratic function, written in the form y = ax^2 + bx + c, is a point on the parabola that represents either the highest or lowest point, depending on whether the parabola opens upwards or downwards. In other words, for this question, the task is to calculate the coordinates (x, y) of the vertex of the parabola. This involves using either the vertex formula, completing the square, or the fact that the x-coordinate of the vertex is at -b/(2a), followed by finding the corresponding y-coordinate by plugging this x-value back into the original equation.
$y = - 2 x^{2} - 16 x - 24$
Answer
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 $ax^2 + bx + c$.
$$a = -2, b = -16, c = -24$$
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 $y = ax^2 + bx + c$, 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 $e$.
