Polynomial and Rational Functions

Oct 13, 2025 by windailys.com 40 views

Rewriting a Quadratic Function to Find its Vertex and Sketch its Graph

Introduction

Quadratic functions are a type of polynomial function that can be written in the form $f(x) = ax^2 + bx + c$ . They are commonly used to model real-world situations, such as the trajectory of a projectile or the growth of a population. In this article, we will focus on rewriting a quadratic function in the form $f(x) = a(x-h)^2 + k$ , which is known as the vertex form. This form is useful for finding the vertex of the parabola and sketching its graph.

The Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by:

$f(x) = a(x-h)^2 + k$

where $a$ , $h$ , and $k$ are constants. The vertex of the parabola is located at the point $(h, k)$ .

Rewriting a Quadratic Function in Vertex Form

Consider the following quadratic function:

$f(x) = 2x^2 - 12x + 11$

Our goal is to rewrite this function in the form $f(x) = a(x-h)^2 + k$ . To do this, we need to complete the square.

Completing the Square

To complete the square, we start by factoring out the coefficient of the $x^2$ term, which is 2:

$f(x) = 2(x^2 - 6x) + 11$

Next, we add and subtract the square of half the coefficient of the $x$ term, which is $(-6/2)^2 = 9$ , inside the parentheses:

$f(x) = 2(x^2 - 6x + 9 - 9) + 11$

Now, we can rewrite the expression inside the parentheses as a perfect square:

$f(x) = 2((x - 3)^2 - 9) + 11$

Next, we distribute the 2 to the terms inside the parentheses:

$f(x) = 2(x - 3)^2 - 18 + 11$

Finally, we simplify the expression by combining the constant terms:

$f(x) = 2(x - 3)^2 - 7$

Finding the Vertex

Now that we have rewritten the function in vertex form, we can easily find the vertex of the parabola. The vertex is located at the point $(h, k)$ , where $h$ is the value of $x$ that makes the expression inside the parentheses equal to 0, and $k$ is the value of the function at that point.

In this case, we have:

$h = 3$

To find $k$ , we substitute $x = 3$ into the function:

$k = f(3) = 2(3 - 3)^2 - 7 = -7$

Therefore, the vertex of the parabola is located at the point $(3, -7)$ .

Sketching the Graph

Now that we have found the vertex of the parabola, we can sketch its graph. The graph of a quadratic function is a parabola that opens upward or downward. The vertex of the parabola is the lowest or highest point on the graph.

In this case, the parabola opens upward, and the vertex is located at the point $(3, -7)$ . We can sketch the graph by starting at the vertex and drawing a parabola that opens upward.

Conclusion

In this article, we have shown how to rewrite a quadratic function in the form $f(x) = a(x-h)^2 + k$ , which is known as the vertex form. We have also shown how to find the vertex of the parabola and sketch its graph. This form is useful for modeling real-world situations and for sketching the graph of a quadratic function.

Example 1

Consider the following quadratic function:

$f(x) = x^2 + 4x + 4$

Rewrite the function in the form $f(x) = a(x-h)^2 + k$ .

Solution

To rewrite the function in the form $f(x) = a(x-h)^2 + k$ , we need to complete the square.

$f(x) = (x^2 + 4x + 4) - 4$

Next, we add and subtract the square of half the coefficient of the $x$ term, which is $(-4/2)^2 = 4$ , inside the parentheses:

$f(x) = (x^2 + 4x + 4 - 4) - 4$

Now, we can rewrite the expression inside the parentheses as a perfect square:

$f(x) = (x + 2)^2 - 4$

Therefore, the function can be rewritten in the form $f(x) = a(x-h)^2 + k$ as:

$f(x) = (x + 2)^2 - 4$

Example 2

Consider the following quadratic function:

$f(x) = 3x^2 - 12x + 15$

Rewrite the function in the form $f(x) = a(x-h)^2 + k$ .

Solution

To rewrite the function in the form $f(x) = a(x-h)^2 + k$ , we need to complete the square.

$f(x) = 3(x^2 - 4x) + 15$

Next, we add and subtract the square of half the coefficient of the $x$ term, which is $(-4/2)^2 = 4$ , inside the parentheses:

$f(x) = 3(x^2 - 4x + 4 - 4) + 15$

Now, we can rewrite the expression inside the parentheses as a perfect square:

$f(x) = 3((x - 2)^2 - 4) + 15$

Next, we distribute the 3 to the terms inside the parentheses:

$f(x) = 3(x - 2)^2 - 12 + 15$

Finally, we simplify the expression by combining the constant terms:

$f(x) = 3(x - 2)^2 + 3$

Therefore, the function can be rewritten in the form $f(x) = a(x-h)^2 + k$ as:

$f(x) = 3(x - 2)^2 + 3$

Conclusion

Final Thoughts

Quadratic functions are a type of polynomial function that can be written in the form $f(x) = ax^2 + bx + c$ . They are commonly used to model real-world situations, such as the trajectory of a projectile or the growth of a population. In this article, we have shown how to rewrite a quadratic function in the form $f(x) = a(x-h)^2 + k$ , which is known as the vertex form. We have also shown how to find the vertex of the parabola and sketch its graph. This form is useful for modeling real-world situations and for sketching the graph of a quadratic function.

References

[1] "Quadratic Functions" by Math Open Reference
[2] "Vertex Form of a Quadratic Function" by Purplemath
[3] "Completing the Square" by Math Is Fun
Polynomial and Rational Functions =====================================

Q&A: Polynomial and Rational Functions

Introduction

In this article, we will answer some of the most frequently asked questions about polynomial and rational functions. These functions are a type of mathematical function that can be used to model real-world situations, such as the trajectory of a projectile or the growth of a population.

Q1: What is a polynomial function?

A polynomial function is a type of mathematical function that can be written in the form:

$f(x) = ax^n + bx^{n-1} + ... + cx + d$

where $a$ , $b$ , $c$ , and $d$ are constants, and $n$ is a positive integer.

Q2: What is a rational function?

A rational function is a type of mathematical function that can be written in the form:

$f(x) = \frac{p(x)}{q(x)}$

where $p(x)$ and $q(x)$ are polynomial functions.

Q3: How do I determine the degree of a polynomial function?

The degree of a polynomial function is the highest power of the variable $x$ in the function. For example, the degree of the function $f(x) = 2x^3 + 3x^2 - 4x + 1$ is 3.

Q4: How do I determine the domain of a rational function?

The domain of a rational function is the set of all values of $x$ for which the function is defined. For example, the domain of the function $f(x) = \frac{1}{x}$ is all real numbers except $x = 0$ .

Q5: How do I find the vertex of a parabola?

The vertex of a parabola is the point at which the parabola changes direction. To find the vertex of a parabola, you can use the formula:

$h = -\frac{b}{2a}$

where $a$ and $b$ are the coefficients of the quadratic function.

Q6: How do I sketch the graph of a polynomial function?

To sketch the graph of a polynomial function, you can use the following steps:

Determine the degree of the function.
Determine the leading coefficient of the function.
Determine the x-intercepts of the function.
Determine the y-intercept of the function.
Use the x-intercepts and y-intercept to sketch the graph of the function.

Q7: How do I find the zeros of a polynomial function?

The zeros of a polynomial function are the values of $x$ for which the function is equal to zero. To find the zeros of a polynomial function, you can use the following steps:

Factor the polynomial function.
Set each factor equal to zero and solve for $x$ .
The values of $x$ that you find are the zeros of the function.

Q8: How do I determine the end behavior of a polynomial function?

The end behavior of a polynomial function is the behavior of the function as $x$ approaches positive or negative infinity. To determine the end behavior of a polynomial function, you can use the following steps:

Determine the degree of the function.
Determine the leading coefficient of the function.
If the degree of the function is even, the end behavior is determined by the sign of the leading coefficient.
If the degree of the function is odd, the end behavior is determined by the sign of the leading coefficient.

Q9: How do I find the horizontal asymptote of a rational function?

The horizontal asymptote of a rational function is the horizontal line that the function approaches as $x$ approaches positive or negative infinity. To find the horizontal asymptote of a rational function, you can use the following steps:

Determine the degrees of the numerator and denominator of the function.
If the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients.
If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$ .
If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Q10: How do I find the vertical asymptote of a rational function?

The vertical asymptote of a rational function is the vertical line that the function approaches as $x$ approaches a certain value. To find the vertical asymptote of a rational function, you can use the following steps:

Determine the values of $x$ for which the denominator of the function is equal to zero.
The values of $x$ that you find are the vertical asymptotes of the function.

Conclusion

In this article, we have answered some of the most frequently asked questions about polynomial and rational functions. These functions are a type of mathematical function that can be used to model real-world situations, such as the trajectory of a projectile or the growth of a population. We hope that this article has been helpful in answering your questions about polynomial and rational functions.

References

[1] "Polynomial Functions" by Math Open Reference
[2] "Rational Functions" by Purplemath
[3] "Vertex Form of a Quadratic Function" by Math Is Fun