Chapter 8.5, Problem 57E

To determine

To find: the quadratic function represent by the graph.

Answer to Problem 57E

  $y=-4x^2+20x-24$

Explanation of Solution

Given information: graph

  

To find the quadratic function for given graph, use with 3 points on the curve to form 3 equation, and solve them to get required result,

Here 3 points are given,

Consider A(3,0), B(2,0), C(1,-8), find the quadratic function using system of equations method,

The general form of quadratic function, $y=a{x}^2+bx+c$ .

Substitute A(3,0), in the quadratic function,

  $\begin{array}{l}a{\left(3\right)}^2+b\left(3\right)+c=0\\ 9a+3b+c=0\mathrm{...}(1)\end{array}$

Substitute B(2,0), in the quadratic function,

  $\begin{array}{l}a{\left(2\right)}^2+b\left(2\right)+c=0\\ 4a+2b+c=0\mathrm{...}(2)\end{array}$

Substitute C(1,-8) in the quadratic function,

  $\begin{array}{l}a{\left(1\right)}^2+b\left(1\right)+c=-8\\ a+b+c=-8\mathrm{...}(3)\end{array}$

Therefore from all the three equation solve the value of a, b, c.

From the equation (1) and (2), we get,

  $a=-4$

And substitute this in first and then from first and third equation,

  $b=20$ and $c=-24$ ,

The value of a, b and c substitute in standard equation of quadratic equation,

  $y=-4x^2+20x-24$

This is the required quadratic function.