Problem 39 - Algebra 2

Question

Sketch each parabola. Identify the axis of symmetry. $$ y=-0.5(x-2)^{2}-5 $$

Step-by-Step Solution

Answer

Vertex at (2, -5), axis of symmetry is the line \(x=2\), and the parabola opens downwards because the leading coefficient is negative.

Step 1: Determine the Vertex

For a parabola in the form of \(y=a(x-h)^{2}+k\), the vertex is located at the point (h, k). Here, the given equation is \(y=-0.5(x-2)^{2}-5\), so the vertex is at (2, -5).

Step 2: Identify the Axis of Symmetry

The axis of symmetry for the parabola is the vertical line that passes through the vertex. In this case, the axis of symmetry is \(x=2\).

Step 3: Determine the Parabola's Direction

The coefficient of \((x-h)^{2}\) determines the direction of the parabola's opening. Since the coefficient is \(-0.5\), which is negative, the parabola opens downwards.

Step 4: Sketch the Parabola

Plot the vertex on the coordinate plane, draw the axis of symmetry, and sketch the parabola opening downwards. Make sure the parabola gets wider as it moves away from the vertex since the coefficient \(-0.5\) is less than 1 in absolute value (but greater than -1), indicating a wider opening than the standard parabola \(y=x^{2}\).

Key Concepts

Axis of Symmetry

Understanding the axis of symmetry is crucial when studying parabolas. It is the imaginary vertical line that divides the parabola into two mirror-image halves. For a standard quadratic equation in the form of y=a(x-h)^2+k, the axis of symmetry is represented by the equation x=h.

As seen in the exercise, the parabola y=-0.5(x-2)^2-5 has its axis of symmetry at x=2. This means if you were to fold the graph of the parabola along this line, both halves would match perfectly. The axis of symmetry also helps us locate the parabola's vertex and is essential in graphing the parabola accurately.

Parabola Vertex

The vertex of a parabola represents the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards. In the context of the quadratic equation y=a(x-h)^2+k, the coordinates of the vertex are given by (h, k).

In our exercise, the parabola y=-0.5(x-2)^2-5 has its vertex at the point (2, -5). The vertex is a pivotal element as it provides a starting point for sketching the parabola. From this point, one could plot additional points equidistant on either side of the axis of symmetry to ensure a symmetric graph. Understanding the importance of the vertex and its location on the coordinate plane is fundamental for accurately drawing parabolas and analyzing their properties.

Direction of Parabola Opening

The direction in which a parabola opens is determined by the sign of the coefficient a in the quadratic equation y=a(x-h)^2+k. If a is positive, the parabola opens upwards; if a is negative, it opens downwards.

In the given equation from our exercise, y=-0.5(x-2)^2-5, the coefficient a is -0.5, indicating that the parabola opens downwards. The value of a also affects the parabola's width; a smaller absolute value of a (less than 1 but greater than 0 or less than 0 but greater than -1) results in a wider parabola. Conversely, a larger absolute value of a will make the parabola narrower. Recognizing the effect of the coefficient on the opening direction helps students not only in sketching parabolas correctly but also in understanding how different equations can give rise to different parabolic shapes.