Question

Complete these steps for the function. a. Tell whether the graph of the function opens up or down. b. Find the coordinates of the vertex. c. Write an equation of the axis of symmetry. $$ y=-5 x^{2} $$

Short answer

The graph of the function opens downwards. The coordinates of the vertex are (0, 0) and the equation for the axis of symmetry is \(x = 0\).

Step-by-step solution

Direction of the Parabola

The sign of \(a\) in the quadratic equation \(y = ax^{2}\) determines the direction of the parabola. If \(a\) is positive, the parabola opens upwards, and if \(a\) is negative, the parabola opens downwards. In this case, \(a = -5\), which is negative. Therefore, the graph of the function opens downwards.

Find the Vertex

For a quadratic function in the form \(y = ax^{2}\), the vertex is always at the origin, (0, 0). This is because the function is symmetric with respect to the y-axis.

Equation of the Axis of Symmetry

The axis of symmetry for a parabola rooted at the origin is always the y-axis, which is the vertical line \(x = 0\).