Chapter 3, Problem 1CC

(a) What is the degree of a quadratic function f?What is the standard form of a quadratic function? How do you put a quadratic function into standard form?
(b) The quadratic function $f(x)=a{(x-h)}^2+k$ is in standard form. The graph of f is a parabola. What is the vertex of the graph of f? How do you determine whether f(h) = k is a minimum or a maximum value?
(c) Express $f(x)=x^2+4x+1$ in standard form. Find the vertex of the graph and the maximum or minimum value of f.

To determine

(a)

To write:

The degree of the quadratic function $f$, the standard form of the quadratic function $f$ and the way to put a quadratic function into standard form.

Answer to Problem 1CC

The degree of the quadratic function $f$ is $2$, the standard form of the quadratic function $f$ is $a{\left(x-h\right)}^2+k$

Explanation of Solution

Calculation:

The quadratic function is $a{x}^2+bx+c$.

The degree of the polynomial is the highest power of the variable in the function.

In quadratic equation the highest power of the variable is $2$.

Therefore, the degree of the quadratic function is $2$.

The standard form of the quadratic function is,

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

By the use of completing the square method the quadratic equation can be put into standard form.

For example apolynomial $P\left(x\right)=x^2+2x+1$.

Solve the quadratic equation $P\left(x\right)=x^2+2x+1$ by the use of completing the square.

$\begin{array}{c}P\left(x\right)=x^2+2x+1\\ =x^2+x+x+1\\ =\left(x+1\right)\left(x+1\right)\\ ={\left(x+1\right)}^2+0\end{array}$

Conclusion:

Hence, the degree of the quadratic function $f$ is $2$, the standard form of the quadratic function $f$ is $a{\left(x-h\right)}^2+k$.