Chapter 3, Problem 21RE

21. In Problems 20 and 21, find the quadratic function for which: Contains the points $\left(0,5\right)$, $\left(1,2\right)$, and $\left(3,2\right)$

To determine

To calculate: The quadratic function from the given points.

Answer to Problem 21RE

The quadratic function is $y=x^2-4x+5$.

Explanation of Solution

Given:

The given points are $(0,5),(1,2)$ and $(3,2)$.

Formula used:

A general quadratic equation is in the form $y=a{x}^2+bx+c$.

We can find the values of $a$, $b$ and $c$ by substituting each point in the general equation and then solving those equations.

Calculation:

Let the equation of the parabola be $y=a{x}^2+bx+c$.

Form the given points, we get that when $x=0$, $y=5$. Thus,

$5=a{(0)}^2+b(0)+c$

$\Rightarrow c=5$-----(1)

When $x=1$, $y=2$. Thus,

$2=a{(1)}^2+b(1)+c$

$\Rightarrow 2=a+b+5$

$\Rightarrow a+b=2-5$

$\Rightarrow a+b=-3$-----(2)

When $x=3$, $y=2$. Thus,

$2=a{(3)}^2+b(3)+c$

$\Rightarrow 2=9a+3b+5$

$\Rightarrow 9a+3b=2-5$

$\Rightarrow 9a+3b=-3$-----(3)

Now, we need to find the values of $a$ and $b$.

We have

$a+b=-3$-----(2)

$9a+3b=-3$-----(3)

On dividing (3) by 3, we get

$3a+b=-1$-----(4)

Now, on subtracting (2) from (4), we get

$2a=2$

$\Rightarrow a=1$

Substituting the value of $a$ in (2), we get

$1+b=-3$

$\Rightarrow b=-3-1=-4$

Thus, we get $a=1,b=-4,c=5$.

Thus, the quadratic equation containing the given 3 points is

$y=x^2-4x+5$