Chapter 9.2, Problem 55SR

To determine

Find the axis of symmetry, coordinate of vertex, graph each function and identify the vertex as a maximum or minimum.

Answer to Problem 55SR

Axis of symmetry is $x=-\frac{1}{3}$

Coordinate of vertex is $(-\frac{1}{3},\frac{2}{3})$

The graph opens up and the vertex is a minimum.

Explanation of Solution

Given:

The quadratic equation: $y=3x^2+2x+1$

Concept Used:

A parabola is a curve where any point is at an equal distance from: a fixed point (the focus), and a fixed straight line (the directrix)

The axis of symmetry (goes through the focus, at right angles to the directrix) The vertex (where the parabola makes its sharpest turn) is halfway between the focus and directrix.

Calculation:

$\begin{array}{l}y=3x^2+2x+1\\ a=3;b=2,\\ x=-\frac{b}{2a}=-\frac{3}{6}=-\frac{1}{3}\end{array}$ Axis of Symmetry: $x=-\frac{1}{3}$ Find the coordinate of the vertex:   $\begin{array}{l}y=3x^2+2x+1\\ $ y=3·{(-\frac{1}{3})}^2+2·(-\frac{1}{3})+1$y=\frac{1}{3}-\frac{2}{3}+1=\frac{1}{3}-\frac{2}{3}+\frac{3}{3}=\frac{2}{3}\end{array}$ The coordinate of the vertex is $(-\frac{1}{3},\frac{2}{3})$. The equation is in standard form and a is positive, the graph opens up and the vertex is a minimum.

Thus, Axis of symmetry is $x=-\frac{1}{3}$ Coordinate of vertex is $(-\frac{1}{3},\frac{2}{3})$ the graph opens up and the vertex is a minimum.