Problem 13 - Algebra 1

Question

Sketch the graph of the function. Label the vertex. $$ y=-5 x^{2}+10 $$

Step-by-Step Solution

Answer

The parabola for the function \(y = -5x^2 + 10\) has a vertex at the point (0, 10) and opens downward.

Step 1: Determine the vertex

The vertex of the function \(y = ax^2 + c\) can be found at the point \(h, k\), where \(h = -\frac{b}{2a}\), and \(k = ac - \frac{b^2}{4a}\). In this case, \(a = -5\) and \(c = 10\), and since there is no x term, \(b = 0\). Plugging these values in gives a vertex point of \((0, 10)\).

Step 2: Determine the direction of the parabola

The coefficient of the \(x^2\) term will determine which way the parabola 'opens'. If \(a > 0\), it opens upward, if \(a < 0\), it opens downward. In this case, \(a = -5\), so the parabola for the function \(y = -5x^2 + 10\) opens downward.

Step 3: Sketch the graph

Start by marking the vertex point (0, 10) on the graph. This is the maximum point of the parabola, since it opens downward. Given the symmetry of parabolas, the y-intercept will also be the vertex in this case. Then sketch a parabola opening downwards from that point.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial point as it marks the peak or the trough of the curve, depending on whether the parabola opens upwards or downwards. In the quadratic function, represented as y = ax^2 + bx + c, the vertex can be calculated using the formulae h = -b/(2a) and k = c - b^2/(4a), where (h, k) are the coordinates of the vertex. If the quadratic equation lacks the x-term like y=-5x^2+10, simplifying the process as b = 0. Therefore, the vertex is at (0, c), which in this instance is (0, 10).

This is an essential concept as the vertex provides not only the maximum or minimum value of the quadratic function but also helps to determine the axis of symmetry and the initial step towards sketching the graph of the parabola.

Direction of a Parabola

Understanding the direction in which a parabola opens is integral in graphing quadratic functions. The sign of the coefficient a in the quadratic function ax^2 will ultimately decide this direction. When a > 0, the parabola's arms extend upwards; conversely, if a < 0 the parabola opens downwards. For the provided exercise with the quadratic function y=-5x^2+10, since a = -5 is negative, the parabola will open downwards. This key piece of information implies that the vertex, located at (0, 10), is the parabola's highest point on the graph. Recognizing the direction is a crucial step not only for sketching the parabola but also for solving many problems related to the maximum or minimum values of a function.

Parabola Symmetry

Parabolas exhibit a significant feature called symmetry about a vertical line known as the axis of symmetry. This axis is a straight line that runs through the vertex and divides the parabola into two mirror-image sections. In any parabola represented by y = ax^2 + bx + c, the axis of symmetry's equation is x = -b/(2a). When b is zero, the axis of symmetry is simply x = 0, which is also the y-axis. In the equation y=-5x^2+10, the symmetry is clearly reflected across the y-axis. Any point on one side of the parabola is mirrored across the axis of symmetry by a corresponding point on the other side. The concept of parabolic symmetry aids in plotting points effectively while graphing the function; once points on one side of the vertex are calculated and plotted, their symmetric counterparts on the other side are easily determined.