Problem 77
Question
The vertex of the graph of \(y=-x^{2}-16 x-62\) lies in which quadrant? \(\mathrm{F} . \mathrm{IV}\) G. III H. II J. 1
Step-by-Step Solution
Verified Answer
The vertex of the graph lies in Quadrant IV.
1Step 1: Determine the vertex of the parabola
The vertex form of a parabola is given by the equation \(y=a(x-h)^{2}+k\), where \(h\) and \(k\) are the x and y coordinates of the vertex, respectively. In the given equation \(y=-x^{2}-16x-62\), we can find the vertex by completing the square or using the vertex formula \(h = -\frac{b}{2a}\). Given that the coefficient \(a=-1\) and \(b=-16\), we can compute the x-coordinate of the vertex as follows: \(h = -\frac{-16}{2(-1)} = -\frac{-16}{-2} = 8\).
2Step 2: Compute the y-coordinate of the vertex
To find the y-coordinate of the vertex, plug the value of \(h\) into the original equation: \(y=-x^{2}-16x-62\). Substituting \(x=h=8\), we get \(y=-(8)^{2}-16(8)-62\). Now, perform the calculations \(y=-64-128-62=-254\). Thus, the y-coordinate of the vertex is -254, so the vertex is \( (8, -254) \).
3Step 3: Determine the quadrant in which the vertex lies
Given the vertex coordinates \( (8, -254) \), where the x-coordinate is positive and the y-coordinate is negative, the vertex lies in the fourth quadrant (Quadrant IV) where \(x>0\) and \(y<0\).
Key Concepts
Vertex of a ParabolaCompleting the SquareCoordinate Geometry
Vertex of a Parabola
Understanding the vertex of a parabola is crucial when analyzing the graph of a quadratic equation. The vertex represents the highest or lowest point on the parabola, depending on whether it opens upwards or downwards. In the given exercise, we have a parabola represented by the quadratic equation \(y=-x^{2}-16x-62\).
To find the vertex, we can use two methods: completing the square or the vertex formula. The vertex formula, \(h = -\frac{b}{2a}\), is derived from the standard form of a quadratic equation, and it provides a quick way to find the x-coordinate of the vertex, referred to as \(h\). In our case, by applying this formula with \(a=-1\) and \(b=-16\), we calculated the x-coordinate to be 8. Plugging this value back into the equation allows us to find the y-coordinate, which results in -254. Consequently, the coordinates of the vertex are \((8, -254)\).
As the parabola opens downwards (indicated by the negative coefficient of \(x^2\)), this vertex is the parabola's highest point. Knowing the vertex is also essential for graphing the parabola, as it serves as a reference point from which the shape of the graph can be sketched.
To find the vertex, we can use two methods: completing the square or the vertex formula. The vertex formula, \(h = -\frac{b}{2a}\), is derived from the standard form of a quadratic equation, and it provides a quick way to find the x-coordinate of the vertex, referred to as \(h\). In our case, by applying this formula with \(a=-1\) and \(b=-16\), we calculated the x-coordinate to be 8. Plugging this value back into the equation allows us to find the y-coordinate, which results in -254. Consequently, the coordinates of the vertex are \((8, -254)\).
As the parabola opens downwards (indicated by the negative coefficient of \(x^2\)), this vertex is the parabola's highest point. Knowing the vertex is also essential for graphing the parabola, as it serves as a reference point from which the shape of the graph can be sketched.
Completing the Square
Completing the square is a method used to transform a quadratic equation into its vertex form, which provides a clearer picture of the graph's characteristics, including the vertex. For the equation \(y=ax^2+bx+c\), completing the square involves creating a perfect square trinomial on the right-hand side of the equation.
To complete the square for the given problem, we would start with the quadratic term and the linear term: \(-x^2 - 16x\). The goal is to add and subtract a particular value that forms a perfect square trinomial, which, in this case, is \((x + 8)^2\). After this process, the equation would be rewritten in the vertex form as \(y = a(x-h)^2 + k\), directly revealing the vertex as \((h, k)\). Although in the given exercise, we utilized the vertex formula for efficiency, completing the square remains a foundational technique for understanding the structure of quadratic equations and their graphs.
To complete the square for the given problem, we would start with the quadratic term and the linear term: \(-x^2 - 16x\). The goal is to add and subtract a particular value that forms a perfect square trinomial, which, in this case, is \((x + 8)^2\). After this process, the equation would be rewritten in the vertex form as \(y = a(x-h)^2 + k\), directly revealing the vertex as \((h, k)\). Although in the given exercise, we utilized the vertex formula for efficiency, completing the square remains a foundational technique for understanding the structure of quadratic equations and their graphs.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, involves the study of geometric figures using the coordinate plane. This system is crucial for locating points, lines, and curves - including parabolas - within a two-dimensional space. Each point is identified by an ordered pair of numbers, \((x, y)\), representing its coordinates along the horizontal (x-axis) and vertical (y-axis) axes.
In coordinate geometry, the concept of quadrants is used to classify the position of points. There are four quadrants in the Cartesian coordinate system, each distinguished by the sign of the x and y coordinates. For instance, a point with a positive x-coordinate and a negative y-coordinate lies in the fourth quadrant. In the exercise, we identified the vertex of the parabola to be at \((8, -254)\), which falls in the fourth quadrant because the x-coordinate (8) is positive, and the y-coordinate (-254) is negative. Understanding the arrangement of quadrants and the signs of the coordinates is fundamental for accurately describing the location of points and shapes within the plane.
In coordinate geometry, the concept of quadrants is used to classify the position of points. There are four quadrants in the Cartesian coordinate system, each distinguished by the sign of the x and y coordinates. For instance, a point with a positive x-coordinate and a negative y-coordinate lies in the fourth quadrant. In the exercise, we identified the vertex of the parabola to be at \((8, -254)\), which falls in the fourth quadrant because the x-coordinate (8) is positive, and the y-coordinate (-254) is negative. Understanding the arrangement of quadrants and the signs of the coordinates is fundamental for accurately describing the location of points and shapes within the plane.
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