Chapter 12.4, Problem 12MRE

To determine

To find: the vertex and the axis of symmetry of the graph of the given equation.

Answer to Problem 12MRE

The coordinate of the vertex is: $\left(\frac{3}{4},-\frac{9}{4}\right)$ .

The axis of symmetry is $x=\frac{3}{4}$ .

Explanation of Solution

Given information:

Given equation,

  $y=-2x^2+3x$

Concept used:

Vertex of parabola of the form $y=f\left(x\right)=a{x}^2+bx+c$ is given by

  $\left(-\frac{b}{2a},f\left(-\frac{b}{2a}\right)\right)$

The equation of axis of symmetry:

  $x=-\frac{b}{2a}$

Calculation:

Consider the given equation:

  $y=-2x^2+3x$

  $x$ Coordinate of the vertex:

  $\begin{array}{c}x=-\frac{3}{2\left(-2\right)}\\ =\frac{3}{4}\end{array}$

To find out the $y$ coordinate, put $x=\frac{3}{4}$ in the given function:

  $\begin{array}{c}y=-2x^2+3x\\ =-2{\left(\frac{3}{4}\right)}^2+3\left(\frac{3}{4}\right)\\ =-2\cdot \frac{9}{4}+3\cdot \frac{3}{4}\\ =-\frac{9}{4}\end{array}$

Hence, the coordinate of the vertex is: $\left(\frac{3}{4},-\frac{9}{4}\right)$ .

The equation of axis of symmetry:

  $\begin{array}{c}x=-\frac{3}{2\left(-2\right)}\\ =\frac{3}{4}\end{array}$

Hence, the axis of symmetry is $x=\frac{3}{4}$ .