Chapter 8, Problem 28CR

To determine

To write:A quadratic function in standard form whose graph passes through points $\left(4,0\right)$ and $\left(6,0\right)$ .

Answer to Problem 28CR

The standard form of a quadratic functionpassing through points through points $\left(4,0\right)$ and $\left(6,0\right)$ is $f\left(x\right)=a\left(x^2-10x+24\right)$ .

Explanation of Solution

Given information:

The graph of quadratic function passes through points $\left(4,0\right)$ and $\left(6,0\right)$ .

Formulaused:

The standard form of a quadratic function presents the function in the form $f\left(x\right)=a{\left(x-h\right)}^2+k$ where $\left(h,k\right)$ is the vertex.

Calculations:

The graph of quadratic function passes through points $\left(4,0\right)$ and $\left(6,0\right)$ .Therefore, $f\left(4\right)=0$ and $f\left(6\right)=0$ .

Now, $f\left(4\right)=0$

  $\Rightarrow a{\left(4-h\right)}^2+k=0$.........(1)

  $f\left(6\right)=0$

  $\Rightarrow a{\left(6-h\right)}^2+k=0$.........(2)

From results (1) and (2)

  $\Rightarrow a{\left(4-h\right)}^2+k=a{\left(6-h\right)}^2+k$

  $\Rightarrow {\left(4-h\right)}^2={\left(6-h\right)}^2$

  $\Rightarrow 16-8h+h^2=36-12h+h^2$

  $\Rightarrow 4h=20$

  $\Rightarrow h=5$.........(3)

Substituting value of h in result (1)

  $a{\left(4-5\right)}^2+k=0$

  $a+k=0$

  $k=-a$.

Now, $f\left(x\right)=a{\left(x-h\right)}^2+k$

  $f\left(x\right)=a{\left(x-5\right)}^2-a$

  $f\left(x\right)=a\left[{\left(x-5\right)}^2-1\right]$

  $f\left(x\right)=a\left(x^2-10x+24\right)$

Hence, the standard form of a quadratic functionpassing through points through points $\left(4,0\right)$ and $\left(6,0\right)$ is $f\left(x\right)=a\left(x^2-10x+24\right)$ , where $a$ is constant is. Since, a can have infinite number of values (real numbers) hence the number of quadratic functions passing through is infinite.

Conclusion:

The standard form of a quadratic functionpassing through points through points $\left(4,0\right)$ and $\left(6,0\right)$ is $f\left(x\right)=a\left(x^2-10x+24\right)$ .