Question

STATEMENT-1: The curve \(y=\frac{-x^{2}}{2}+x+1\) is symmetric with respect to the line \(x=1 .\) because STATEMENT-2 : A parabola is symmetric about its axis. (A) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1 (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement- 2 is False (D) Statement-1 is False, Statement- 2 is True

Short answer

(A) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1

Step-by-step solution

Analyzing the parabola equation

First, rewrite the quadratic equation for the parabola in standard form to identify its vertex. The standard form is given by the equation: \(y = a(x-h)^2 + k\), where \((h, k)\) is the vertex of the parabola. In this case, the equation is \(y = -\frac{1}{2}x^2 + x + 1\). Completing the square will find the vertex form.

Completing the square

To complete the square for the parabola equation \(y = -\frac{1}{2}x^2 + x + 1\), factor out the coefficient of \(x^2\) from the first two terms: \(y = -\frac{1}{2}(x^2 - 2x) + 1\). Then add and subtract \((\frac{2}{2*\sqrt{2}})^2\), so that the first three terms form a perfect square trinomial: \(y = -\frac{1}{2}[(x - 1)^2 - (1)^2] + 1\), which simplifies to \(y = -\frac{1}{2}(x - 1)^2 + \frac{3}{2}\).

Finding the axis of symmetry

From the completed square form \(y = -\frac{1}{2}(x - 1)^2 + \frac{3}{2}\), we can see that the axis of symmetry is the line \(x = 1\) because it is the line that would divide the parabola into two mirrored halves. Thus, statement-1 is true.

Assessing the correctness of statement-2

Statement-2 states that a parabola is symmetric about its axis. This is a standard property of parabolas and is indeed true. However, to determine whether it is the correct explanation for statement-1, we assess if the fact that parabolas are symmetric about their axis directly explains why this particular parabola is symmetric about the line \(x = 1\).

Determine the correct explanation

Since we established that the axis of symmetry for the given parabola is \(x = 1\), and statement-2 correctly describes a general property of parabolas, which directly applies to the given parabola, statement-2 is indeed a correct explanation for statement-1.

Key concepts

Parabola Axis of Symmetry

When studying parabolas, one crucial concept is the axis of symmetry. This imaginary line runs through the parabola and divides it into two mirror-image halves. Identifying the axis of symmetry is vital for understanding the parabola's overall shape and its reflective properties.

Let's consider the quadratic function given in the exercise: \(y=-\frac{x^2}{2}+x+1\). By completing the square, which we'll discuss in more detail shortly, we can rewrite this in vertex form to identify the vertex. From there, the axis of symmetry is directly given by the x-coordinate of this vertex. In our example, the axis of symmetry is \(x=1\), which is why the given curve is symmetric with respect to this line.

The axis of symmetry can be used to find the parabola's maximum or minimum values since they occur at the vertex, which lies on this axis. Understanding this concept aids in graphing parabolas and in solving practical problems such as projectile motion or optics, where symmetry plays a key role.

Completing the Square

Completing the square is an algebraic maneuver used to rewrite quadratic equations such as \(ax^2+bx+c\) in a form that makes the properties of the parabola—such as its vertex, axis of symmetry, and maximum or minimum value—much clearer. This technique involves creating a perfect square trinomial from the quadratic terms of the equation, which can then be expressed as the square of a binomial.

To complete the square for the given function \(y=-\frac{1}{2}x^2+x+1\), we focus on the quadratic and linear terms. For our function, we would add and subtract \((\frac{1}{2})^2\) inside the parentheses, which results in the perfect square trinomial \((x-1)^2\). After simplifying, we get the vertex form: \(y=-\frac{1}{2}(x-1)^2+\frac{3}{2}\). This form quickly reveals the vertex and therefore the axis of symmetry, without needing extra steps or additional calculations. It's a powerful tool in the algebraic toolkit that facilitates a deeper understanding of quadratic functions.

Vertex of a Parabola

The vertex of a parabola is the point where the curve changes direction. It represents either the highest point on the graph (for a downward-opening parabola) or the lowest point (for an upward-opening parabola). The coordinates of the vertex are immensely useful, as they give us the precise location of these turning points.

In the exercise above, after completing the square, we determined that the vertex form of our parabola was \(y=-\frac{1}{2}(x-1)^2+\frac{3}{2}\). In this form, the coordinates of the vertex are directly read as \(h\) and \(k\) from the expression \(a(x-h)^2+k\). So, the vertex for our given parabola is at \((1,\frac{3}{2})\). This particular point is pivotal in understanding the graph's behavior because it acts as the 'apex' from which the parabola extends and shapes itself on the Cartesian plane.

The concept of the vertex is not only fundamental in graphing parabolas but also in various applications of mathematics, including physics, engineering, and economics, where parabolic relationships model diverse phenomena.