Chapter 8, Problem 11CT

To determine

Tofind:The quadratic function in standard form whose graph satisfies the given conditions.

Answer to Problem 11CT

The standard form of quadratic function that passes through points $\left(0,0\right)$ , $\left(10,0\right)$ , and $\left(9,-27\right)$ is

  $f\left(x\right)=3{\left(x-5\right)}^2-75$ .

Explanation of Solution

Given information:

The graph of quadratic function passes through points $\left(0,0\right)$ , $\left(10,0\right)$ , and $\left(9,-27\right)$ .

Formulaused:

The standard form of quadratic function is written as $f\left(x\right)=a{\left(x-h\right)}^2+k$ having vertex at point $\left(h,k\right)$ . A quadratic function $g\left(x\right)$ having $x=\alpha$ , $x=\beta$ as its zeros may be written as $g\left(x\right)=a\left(x-\alpha \right)\left(x-\beta \right)$ , where“a” is a number.

Calculations:

Let $f\left(x\right)$ be the quadratic function that passes through points $\left(0,0\right)$ , $\left(10,0\right)$ , and $\left(9,-27\right)$ .

From the given information we notice that points $\left(0,0\right)$ and $\left(10,0\right)$ are the intercepts of the function $f\left(x\right)$ , i.e., $f\left(0\right)=0$ and $f\left(10\right)=0$ .

Thus, $x=0$ and $x=10$ are the zeros of function $f\left(x\right)$ . Therefore the function $f\left(x\right)$ is $f\left(x\right)=a\left(x-0\right)\left(x-10\right)$

  $f\left(x\right)=ax\left(x-10\right)$.........(1)

Further, the quadratic function $f\left(x\right)$ passes through point $\left(9,-27\right)$ , i.e., $f\left(9\right)=-27$

  $\Rightarrow f\left(9\right)=a\cdot 9\left(9-10\right)$

  $\Rightarrow -27=a\cdot 9\left(-1\right)$

  $\Rightarrow a=\frac{-27}{-9}=3$ .

Thus, the function $f\left(x\right)=3x\left(x-10\right)$ or $f\left(x\right)=3\left(x^2-10x\right)$ . This, is general form of required quadratic function.

Now, $f\left(x\right)=3\left(x^2-10x+25-25\right)$

  $f\left(x\right)=3\left(x^2-10x+25\right)-75$

  $f\left(x\right)=3{\left(x-5\right)}^2-75$.........(2)

Result (2) is the standard form of quadratic function that passes through points $\left(0,0\right)$ , $\left(10,0\right)$ , and $\left(9,-27\right)$ .

Conclusion:

The standard form of quadratic function that passes through points $\left(0,0\right)$ , $\left(10,0\right)$ , and $\left(9,-27\right)$ is

  $f\left(x\right)=3{\left(x-5\right)}^2-75$ .