Chapter 3, Problem 1RCC

1. (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?
2. (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?
3. (c) Express f(x) = x² + 4x + 1 in standard form. Find the vertex of the graph and the maximum or minimum value of f.

To determine

To find: The degree of quadratic function f and its standard form and the method to put a quadratic function into standard form.

Answer to Problem 1RCC

The degree of the quadratic function $f\left(x\right)$ is two, the standard form of the quadratic function is $f\left(x\right)=a{\left(x-h\right)}^2+k$ and the quadratic expression can be expressed in standard form by completing the square.

Explanation of Solution

Calculation:

The general form of a quadratic function is, $f\left(x\right)=a{x}^2+bx+c$.

Where, a, b and c represents a real number.

And the degree of the quadratic function $f\left(x\right)$ is two.

That is, quadratic function f is a polynomial function of second degree.

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

And the parabola opens upward if $a>0$ or downward if $a<0$.

Quadratic expression can be expressed in standard form by completing the square.

Assume a quadratic function $y=3x^2-6x+4$.

Convert the quadratic function $y=3x^2-6x+4$ in standard form as follows,

$\begin{array}{c}y=3x^2-6x+4\\ y-4=3x^2-6x\\ y-4=3\left(x^2-2x\right)\\ y-4+3=3\left(x^2-2x\right)+3\\ y-4+3=3\left(x^2-2x+1\right)\end{array}$

Simplify further as follows,

$\begin{array}{c}y-1=3{\left(x-1\right)}^2\\ y=3{\left(x-1\right)}^2+1\end{array}$

Thus, the quadratic function $y=3x^2-6x+4$ in standard form is, $y=3{\left(x-1\right)}^2+1$.

Therefore, the degree of the quadratic function $f\left(x\right)$ is two, the standard form of the quadratic function is $f\left(x\right)=a{\left(x-h\right)}^2+k$ and the quadratic expression can be expressed in standard form by completing the square.

To determine

To explain: The vertex of the graph $f\left(x\right)=a{\left(x-h\right)}^2+k$ and $f\left(h\right)=k$ is minimum or maximum value.

Explanation of Solution

The given function is, $f\left(x\right)=a{\left(x-h\right)}^2+k$.

It is in the form of parabola.

Thus, the vertex of the parabola is $\left(h,k\right)$.

The parabola opens upward if $a>0$ or downward if $a<0$.

Thus, if $a<0$ then $f\left(h\right)$ is the minimum value of the function and if $a>0$ then $f\left(h\right)$ is the maximum value of the function.

To determine

To find: The standard form of $f\left(x\right)=x^2+4x+1$ also the vertex of the graph and the minimum or maximum value of f.

Answer to Problem 1RCC

The standard form of the function $f\left(x\right)=x^2+4x+1$ is ${\left(x+2\right)}^2-3$.

The maximum value of the expression f is $-3$.

Explanation of Solution

Given:

The given equation is, $f\left(x\right)=x^2+4x+1$.

Formula used:

Maximum or minimum value of a quadratic function:

The standard form of a quadratic function $f\left(x\right)=a{x}^2+bx+c$ is $f\left(x\right)=a{\left(x-h\right)}^2+k$. The maximum or minimum value of $f\left(x\right)$ occurs at $x=h$.

If $a>0$, then the minimum value of f is $f\left(h\right)=k$.

If $a<0$, then the maximum value of f is $f\left(h\right)=k$.

Calculation:

The quadratic expression can be expressed in standard form by completing the square.

Convert the given quadratic function in standard form as follows,

$\begin{array}{c}f\left(x\right)=x^2+4x+1\\ f\left(x\right)-1=x^2+4x\\ f\left(x\right)-1+4=x^2+4x+4\\ f\left(x\right)+3={\left(x+2\right)}^2\end{array}$

$f\left(x\right)={\left(x+2\right)}^2-3$

Thus, the quadratic function in standard form is ${\left(x+2\right)}^2-3$.

Compare this equation with the standard form $f\left(x\right)=a{\left(x-h\right)}^2+k$.

Where, $a=1,h=-2$ and $k=-3$

The vertex of the parabola is $\left(h,k\right)$.

So the vertex of the graph of the given function is $\left(-2,3\right)$.

From the above formula, it is known that minimum value exists at $f\left(h\right)=k$ if $a>0$.

The quadratic function $f\left(x\right)=x^2+4x+1$ has the minimum value and it exists at $f\left(-2\right)=-3$.

Therefore, minimum value of the function $f\left(x\right)=x^2+4x+1$ is $-3$.