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. $$ y=-7 x^{2}+2 x $$

Short answer

The graph of the function opens downward. The vertex of the parabola is \((\frac{1}{7}, -\frac{5}{7})\). The equation for the axis of symmetry is \(x=\frac{1}{7}\).

Step-by-step solution

Identify the Opening Direction of the Graph

The coefficient of \(x^2\) in the equation is -7. When the coefficient 'a' is negative, the graph of the function opens downward.

Compute the Vertex of the Parabola

The coordinates of the vertex of a parabola given by \(y=ax^2+bx+c\) are \((-\frac{b}{2a},f(-\frac{b}{2a})\). Here, 'a' is -7 and 'b' is 2, so the x-coordinate of the vertex is \(-\frac{b}{2a}=-\frac{2}{2*(-7)}=\frac{1}{7}\). Substitute \(x =\frac{1}{7}\) into the equation to find the y-coordinate of the vertex, which comes out to be \(y=-7*(\frac{1}{7})^2+2*(\frac{1}{7})=-1+\frac{2}{7}= -\frac{5}{7}\). Therefore, the vertex of the parabola is \((\frac{1}{7},-\frac{5}{7})\)

Write an Equation for the Axis of Symmetry

The equation for the axis of symmetry of a parabola given by \(y=ax^2+bx+c\) is \(x=-\frac{b}{2a}\). Substituting 'a' and 'b' with -7 and 2 respectively, the equation for the axis of symmetry of this parabola is \(x=\frac{1}{7}\).